ON THE SECOND RATIONAL K-GROUP OF AN ELLIPTICCURVE OVER GLOBAL FIELDS OF POSITIVE
CHARACTERISTICSATOSHI KONDO AND SEIDAI YASUDA

Abstract. Let E be an elliptic curve over a global ﬁeld of positive characteristic. Let r be the order of zero at s = 0 of the Hasse-Weil L-function with bad
factors removed. The Parshin conjecture on the vanishing of higher rational Ktheory of projective smooth schemes over ﬁnite ﬁelds implies dimQ K2 (E) ⊗Z
Q = r. It is shown that dimQ K2 (E) ⊗Z Q ≥ r.

(This preprint is a revised version of the ﬁrst half of our preprint RIMS-1564,
RIMS, Kyoto University, August 2006.)
1. Introduction
Let C be a connected projective smooth curve over a ﬁnite ﬁeld. Let E → C
be an elliptic surface, that is, a proper, ﬂat, generically smooth morphism from a
regular scheme E to C such that for almost all closed points ℘ of C, the ﬁber E ×C ℘is a genus 1 curve. Let E denote the generic ﬁber of E.
For a closed point ℘ of C, let κ(℘) be the residue ﬁeld at ℘ and E℘ = E ×C
Spec κ(℘) be the ﬁber at ℘. Let us identify the K-theory and the G-theory of
regular noetherian schemes. There is a localization sequence of G-theory:⊕∂℘ M
K2 (E) → K2 (E) −−−→
G1 (E℘ ) → K1 (E)℘where ℘ runs over all closed points of C.
We use the subscript −Q to mean − ⊗Z Q.
Theorem 1.1. Let the notations be as above. Then the homomorphism⊕∂℘Q M
K2 (E)Q −−−→
G1 (E℘ )Q℘

is surjective.
Let n be a positive integer. The Parshin conjecture says that Kn (X)Q = 0 for
any smooth projective variety X over a ﬁnite ﬁeld. The validity of the conjecture
in particular implies that K1 (E)Q = K2 (E)Q = 0, hence that the homomorphism inTheorem 1.1 is an isomorphism. One can compute exactly the target space when
E is an elliptic curve (see Lemma 2.2).
During this research, the ﬁrst author was supported as a Twenty-First Century COE Kyoto
Mathematics Fellow, by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, and was partially supported by JSPS Grant-in-Aid for Scientiﬁc Research
17740016. The second author was partially supported by JSPS Grant-in-Aid for Scientiﬁc Research 16244120.
1

2

SATOSHI KONDO AND SEIDAI YASUDALet E be an elliptic curve over a global ﬁeld k of positive characteristic. The ﬁrst
author’s principal motivation was to prove the following corollary, which we will
deduce from Theorem 1.1 in Section 11 by taking C to be the projective smooth
curve over a ﬁnite ﬁeld whose function ﬁeld is k and E to be the ﬂat proper regularminimal model of E over C. For a ﬁnite set M , we let |M | denote the cardinalityof M .
Corollary 1.2. Let S be the set of primes of k at which E has split multiplicative
reduction. Then dimQ K2 (E)Q ≥ |S|.
Let LS (E, s) denote the Hasse-Weil L-function of E with the local factors at
the primes in S removed. Then Corollary 1.2 gives a relation between the rank of
K2 (E) and the L-function LS (E, s), since the order of zero at s = 0 of LS (E, s) isequal to |S|.
The analogue of Corollary 1.2 over number ﬁelds is that the rational rank of theK-group is greater than or equal to |S| + [k : Q] (see [Ro-Sc, Section 1.2, p. 62]). It
is a consequence of (a strong form of) Beilinson’s conjectures, and is not yet proved.
Bloch and Grayson give a method ([Bl-Gr, p. 81-83]) for constructing elements in
K2 such that the image under the boundary Q
map is not trivial, but one requires
the condition that the image of Gal(k/k) → l EndZ` (T` (E(k))) (where k is theseparable closure of k and ` runs over all primes) is small. Hence their method is
not applicable in general.
As an application, we see that there exists an elliptic curve over a global ﬁeld ofpositive characteristic with arbitrarily large rational K2 .
Corollary 1.3. Let k be a global ﬁeld of positive characteristic. Then for any
integer n ≥ 0, there exists an elliptic curve E over k such that dimQ K2 (E)Q ≥ n.
We have another application of Theorem 1.1. From the theorem, we infer thatthe cokernel of ⊕∂℘ is torsion. We show in our forthcoming paper that the torsion
group is a ﬁnite group, and that the order is explicitly determined when E → C is
non-isotrivial and has a section.
To prove Theorem 1.1 we use modular parameterizations of E by Drinfeld modular curves and the analogue of Beilinson elements, constructed in [Ko-Ya], forDrinfeld modular curves. The key idea is that, for every place ℘ at which the elliptic curve has split multiplicative reduction, one has a modular parametrizationof E by a Drinfeld modular curve where ℘ plays the role of the archimedean placeof Q, and hence one has an analogue of Beilinson elements for every such ℘. The
linear independence of the elements amounts to the integrality of the elements.
The sections are organized as follows. Sections 2 and 3 do not depend on othersections. The output are Proposition 2.1, Lemma 2.2 and Proposition 3.1. The
theme of these two sections is to compare the G-group of the special ﬁber (resp.
the boundary map from the generic ﬁber) and the étale cohomology of the genericﬁber (resp. the Chern class map). Sections 4, 5, and 6 compute the moduli ofDrinfeld modular curves. Recall that there is an analogy between the moduli of
elliptic curves and the Drinfeld modular curves. The analogous statements given inthese three sections (see the next paragraphs for the contents) are well known forthe case of elliptic modular curves. We supply the proofs for our Drinfeld modularcase since we were not able to ﬁnd a good reference. The results of these three
sections will be used in Section 7. Sections 7, 8, 9, and 10 contain the Drinfeld
modular analogue of the work of Beilinson ([Be], see also the book [RSS]). We refer

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

3

to the introduction in Section 7 for a very short paragraph on Beilinson’s result.
Section 11 contains the idea which does not have an analogue in the number ﬁeld
case, namely to vary the modular parametrizations to construct (many) elementsin the rational K2 of the elliptic curve.
We give the contents of each Section below. The reader is referred to the beginning of each Section for more technical details.
In Section 2, we compute the target group of the homomorphism ⊕℘ ∂℘Q of Theorem 1.1. If one is interested only in the minimal model, the proof is easy (we refer
to [Bl-Gr, p.82-83]). The reader may read only the statements of Proposition 2.1
and Lemma 2.2 and skip the proof.
Section 3 contains Proposition 3.1. This is a rather technical statement, and theproof is again a lot of diagram chasing in (co)homology theories. We explain how
Proposition 3.1 will be used. The main ingredient for the proof of Theorem 1.1 isLemma 8.1, which is stated in terms of the regulator (which is close to the boundary
map to the special ﬁber). We use Proposition 3.1 to convert it to the statement
solely in terms of generic ﬁber, that is, in a “model free” manner. The reader may
skip the technical details of this section.
In Section 4, we construct what is called the determinant morphism. This is amorphism from the moduli of Drinfeld modules of rank d to that of rank 1. The only
case of our interest is the case d = 2; this may be regarded as the analogue of themorphism obtained from the Weil pairing in the case of ellitpic curves. Although
we do not construct the actual pairing, we mention that the construction of theanalogue of the Weil pairing for Drinfeld modules have been done by Hamahata [Ha]for the rank two Drinfeld modules over the base ring A = Fq [t] using the theory oft-motives, and by van den Heiden [vdHe] for general rank over the general base A
using the theory of A-motives. We also give the details on the uniformization of
the moduli space using elliptic sheaves. See the beginning of the section for more
information.
In Section 5, we give the details on an analogue of Deuring’s theory on supersingular elliptic curves for Drinfeld modules of rank two. The essential part of theresults is due to Drinfeld; there are also papers [Gek4], [Gek5], [Gek3] by Gekeler.
The output of this section are the isomorphism (5.4) and Proposition 5.4. We
describe the set of supersingular points as a double coset space by introducing aquaternion algebra (obtained from the endomorphism ring of a supersingular Drinfeld module). The statements are standard in the case of moduli of elliptic curves;
the reader who is familiar with them may skip the details of this section.
In Section 6, we describe the reduction modulo a prime of Drinfeld modular
curve. Analogous statements for elliptic modular curves are found in the book
[Ka-Ma]. We give the details for we were not able to ﬁnd them in the literature.
In Section 7, we prove the integrality of certain elements in the K2 of Drinfeld
modular curves. This section is the function ﬁeld analogue of [Sc-Sc, Section 7,
p. 298-302].
The aim of Section 8 is to quote a theorem in our other paper [Ko-Ya]. In
the course, we describe the Bruhat-Tits building (which is closely related to theDrinfeld symmetric space) and a “regulator” map (which is a map from the Kgroup of the moduli space to the space of automorphic forms). Lemma 8.1 roughly
says that there exists a subspace in the rational K2 of the Drinfeld modular curve
which maps surjectively to the space of harmonic cochains.

4

SATOSHI KONDO AND SEIDAI YASUDA

In Section 9, we apply Lemma 8.1 and prove Proposition 9.1. This roughly saysthat there exists a certain subspace in the rational K2 of the compactiﬁed Drinfeld
modular curve.
In Section 10, we recall the modular parametrization of an elliptic curve overa function ﬁeld by a (compactiﬁed) Drinfeld modular curve. We apply this (Section 11.1) to construct certain elements in the rational K2 of the elliptic curvefrom the elements in the rational K2 of the compactiﬁed Drinfeld modular curve ofSection 9.
In the ﬁnal Section 11, we give proofs of Theorem 1.1 and Corollaries 1.2 and1.3.
Acknowledgment The ﬁrst author is grateful to Professor Bloch for numerous discussions and to the University of Chicago for hospitality. He also thanks
Masanori Asakura for his interest, Fumiharu Kato for comments on rigid analysis,
and Takeshi Abe for discussions.
The second author would like to thank Teruyoshi Yoshida for discussions aboutbad reductions of modular varieties.
The authors would like to thank Takuya Yamauchi for his suggestion of thestatement of Corollary 1.3, and the referee for his/her comments which improved
our article.
2. The rational G-group of the reduction
The aim of this section is to compute the target space of the homomorphism⊕℘ ∂℘Q in Theorem 1.1. When E is an elliptic curve, i.e., when the canonicalmorphism E → Spec k has a section, the result is Lemma 2.2. This will only be
used in Section 11.1. For the proof of Lemma 2.2, we use Proposition 2.1. This willturn out to be useful in the proof of Lemma 11.2.
The result (Lemma 2.2) is very simple but the proof is long. If one is to assume
that the model E is a minimal model, then the computation is easy, since the rankof the rational G1 is then equal to the rank of the ﬁrst homology of the dual graph(see [Bl-Gr, p.82-83]). The reader interested only in the minimal model case may
skip the entire section.
2.1. We give a proof of the following proposition in Sections 2.1.1-2.1.5.
Let S be the spectrum of a henselian discrete valuation ring whose residue ﬁeld isa ﬁnite ﬁeld of characteristic p. We denote by s (resp. η) the closed (resp. generic)point in S. Let X → S be a proper ﬂat morphism from a regular scheme X to S
such that the generic ﬁber Xη is a smooth curve over η. Let Xs = X ×S s denotethe special ﬁber. Let l be a prime number diﬀerent from p.
Proposition 2.1. Let the notations be as above. We have2
dimQ G1 (Xs )Q = dimQ` Hét
(Xη , Q` (2)).

2.1.1. We let Y = Xsred denote the reduced scheme associated with Xs . It is
known that the pushforward map of the canonical closed immersion Y ,→ Xs∼=→ G1 (Xs ).
induces an isomorphism of G-theory. In particular, we have G1 (Y ) −ji

Let Y sm ,→ Y ←- Y sing denote the smooth locus and the singular locus with thegreduced scheme structure of Y respectively. Let Ye −→ Y denote the normalization.
sm = Y sm × Y
sing = Y sing × Y
e and Y]e.
We put Yg
YYON THE RATIONAL K2 OF AN ELLIPTIC CURVE

5

We have a commutative diagrameejism −−−sing
Yg−→ Ye ←−−−− Y]




g
 singg sm yyygji

Y sm −−−−→ Y ←−−−− Y singwhere the two squares are cartesian. Note that g sm is an isomorphism and that Ye
is a (not necessarily connected) smooth curve.
2.1.2. We obtain a commutative diagram:sm )G1 (YgQ∂Ye/ G (Y]sing )
0
Q

sm
∼
= g∗

(1)

G1 (Y )Q²

/ G1 (Y sm )Q∂Y²g∗sing/ G0 (Y sing )Q

where ∂Ye and ∂Y are the boundary maps in the localization sequence. The restriction map (1) is injective since the G1 of a ﬁnite ﬁeld is torsion. We will computethe dimension of the kernel of ∂Y by comparing with étale Borel-Moore homology.
2.1.3. Let fY : Y → s and fYe : Ye → s denote the structure morphisms.
Let ` be a prime number prime to p. Let F be a constructible étale Q` -sheaf on s.
In the derived category of complexes of étale Q` -sheaves with bounded constructible
cohomology on Y and on Ye , we have distinguished triangles i∗ i! fY! F → fY! F →j∗ j ∗ fY! F →, and ei∗ei! fYe! F → fYe! F → ej∗ej ∗ fYe! F → . Consider the following diagram:fY ∗ i∗ i! fY! F −−−−→ fY ∗ fY! F −−−−→ fY ∗ j∗ j ∗ fY! F
x
x
x







fYe ∗ei∗ei! fYe! F −−−−→ fYe ∗ fYe! F −−−−→ fYe ∗ej∗ej ∗ fYe! F.
The rows are obtained from the triangles above by applying fY ∗ and fYe ∗ respectively. Using base change, we see that fYe ∗ei∗ei! fYe! = fY ∗ i∗ i! g∗ g ! fY! and fYe ∗ej∗ej ∗ fYe! =∗! !
fY ∗ j∗ j g∗ g fY . We deﬁne the vertical arrows to be those induced from the traceg∗ g ! → Id, so the diagram is commutative.
We deﬁne the Borel-Moore étale homology group of a scheme X of ﬁnite type−a!
over s to be HaBM (X, F) = Hét
(X, fXF) where fX is the structure morphism andF is a constructible étale sheaf of Z/ln -modules on s (see [Ja-Sa, p.488, Example
2.2(a)]). As the rows of the diagram above induce the localization sequence and thevertical arrows induce the pushforward maps (see loc. cit.), we obtain the following
commutative diagram:
sm , Q )H1BM (Yg`∼=

H1BM (Y, Q` )(2)²

/ H BM (Y]sing , Q )
`

0

g∗sm

/ H BM (Y sm , Q` )
1

∂BM
g

∂BM

²

g∗sing/ H BM (Y sing , Q` ).
0

1
Since H1BM (Y sing , Q` ) ∼(Y sing , Q` (1)) = 0, the map (2) is injective.
= Hét6

SATOSHI KONDO AND SEIDAI YASUDA

2.1.4. Now we compare G-theory and étale Borel-Moore homology. We consider
the following diagram:
∂Ye

sm )
G1 (YgQO/ G (Y]sing )
0
Q
O
∼
=

=
∂Ye

sm )
G1 (Yg
Q
O

/

∼
=
sm )
K1 (Yg
Q

²

1 g
Hét
(Y sm , Q` (1))

sing
G0 (Y]O∂Ye

sing/ K Y^
(Ye )

Q

0

∂

/ H2

^
sing ,ét
Y²(Ye , Q` (1))

(3)
∼
=

(4)
∼
=

/

/

sing
K0 (Y]²
0 ]Hét(Y sing
∼
=

1

∂BM
g

/

red

)Q

red

²

sing
H0BM (Y], Q` )

red, Q` )

∼
=

=

²
BM g
H1 (Y sm , Q` )

)Q

∼
=

∼
=

²
sm , Q )H BM (Yg
`

red

∂BM
g

²
/ H BM (Y]sing , Q ).
`
0

red

sing
sing . Note that it isHere we let Y]
denote the reduced scheme associated with Y]
a reduced zero dimensional scheme hence regular. In the top commutative rectangle,
the right vertical arrow is the pushforward by the canonical closed immersion.
In the second rectangle, the vertical maps are the canonical isomorphism of
G-groups and K-groups of noetherian regular schemes. The ∂Ye means the bound^sing
ary map in the respective localization sequence. Here K0Y (Ye ) is K-theory withsupport. The rectangle is known to be commutative.
In the third rectangle, the entries of the bottom row (the fourth row of thewhole diagram) are the étale cohomology and étale cohomology with support. The
three vertical arrows are the Chern class maps and the maps (3) and (4) are theisomomorphisms in [Gi, p.235, Theorem 3.1]. It is known that both squares arecommutative ([Gi] loc. cit.). We also know that the ﬁrst and the third verticalarrows induce isomorphisms when the sources are tensored by Q` .
In the fourth rectangle, the vertical maps are the canonical maps induced from!
∗fY!g
= fY∗g[2](1) and f^= f^. The horizontal arrow ∂BM
g is the boundarysmsm
Y singY singmap. The rectangle is commutative.
In the ﬁfth rectangle, the right vertical arrow is the morphismr∗ r ! f ^Q`
red Q` = f ^Y sing ∗Y sing
^
sing
∗ Y
∗
!
= D(f^r∗ r f ^Q` ) → D(f^f^Q` ) = f ^
f^
Q`
sing
sing
Y sing ∗Y singY sing ∗ Y
Y sing ∗ Y
!
f^
red f
sing
Y

induced by the adjunction id → r∗ r∗ . Here D denotes the duality. It follows from
[SGA4-2, p.367, Theorem 1.1] that the morphism is an isomorphism.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

7

Some diagram chasing now yields
dimQ G1 (Y )Q = dimQ` H1BM (Y, Q` ).
2.1.5. To compute H1BM (Y, Q` ), note that we have an isomorphism (see [Ja-Sa,3
p.489, Example 2.2(e)]) H1BM (Y, Q` ) ∼(X, Q` (2)). We use the exact sequence= HY,ét
α

2
2
3
Hét
(X, Q` (2)) → Hét
(Xη , Q` (2)) −→ HY,ét(X, Q` (2))
3
→ Hét (X, Q` (2))
mof étale cohomology groups. Note that by proper base change, we have Hét
(X, Q` (2)) ∼=m
Hét(Y, Q` (2)). It is zero when m = 2 or 3 by the weight argument. Hence α is an
isomorphism. This completes the proof of Proposition 2.1.

2.2. We give an application of Proposition 2.1. Let E → C and E be as in the
setup of Theorem 1.1.
Lemma 2.2. Assume that E → Spec k has a section, so that E is an elliptic curve.
Let S ⊂ C denote the set of places of k at which E has split multiplicative reduction.
Then we have½
1 if ℘ ∈ S,
(2.1)
dimQ G1 (E℘ )Q =0 if ℘ ∈/ S.
Proof. Observe that using Proposition 2.1, the left hand side depends only on
the generic ﬁber, which means that it is independent of the choice of the model.
Suppose E is a minimal model. The reduction of a minimal model is classiﬁed (see[Liu, p.484, 10.2.1]). One can compute directly the Borel-Moore étale homology (orthe G-theory) and see that the claim of the lemma holds true.
¤
3. From special fiber to generic fiber
The aim of this section is to prove Proposition 3.1 below. It compares theboundary map which appeared in the statement of Theorem 1.1 with the Chernclass map. This result will be used only for the proof of Theorem 1.1 in Section 11.
3.1. Let the notations (such as X, S, s, etc.) be as in Section 2. We let Y
denote the complement X \ Xη with the reduced scheme structure. We consider
the boundary map∂Q : K2 (Xη )Q → G1 (Y )Q
supplied by the localization sequence [Qu, p. 127, Proposition 3.2] of the G-theory
for the pair (X, Y ). We ﬁx a prime number ` diﬀerent from p. We consider theétale Chern class map2c2,2,Q : K2 (Xη )Q → Hét
(Xη , Q` (2)),(or more precisely, the limit of étale Chern class maps with ﬁnite coeﬃcients)
introduced in [Gi, p. 228–229, Deﬁnition 2.22].
Proposition 3.1. For every element x ∈ K2 (Xη )Q , we have ∂Q (x) = 0 if and only
if c2,2,Q (x) = 0.
We give two lemmas then the proof.

8

SATOSHI KONDO AND SEIDAI YASUDA

3.2. Let R denote the ring of global sections of the sheaf of total quotient ringsof OY . Let j : G1 (Y )Q → G1 (R)Q = (R× )Q denote the canonical map. Let1c1,1,Q : (R× )Q → Hét(Spec R, Q` (1)) be the homomorphism supplied by the Kummer theoryn1
R× /(R× )` ∼(Spec R, Z/`n (1)).
= HétLemma 3.2. For an element x ∈ K2 (Xη )Q , we have ∂Q (x) = 0 if and only if
c1,1,Q ◦ j ◦ ∂Q (x) = 0.
Proof. Note that, since the residue ﬁeld of a closed point Y is a ﬁnite ﬁeld, thegroup G1 (y) is ﬁnite for each closed point y of Y . This implies that the canonicalhomomorphism j : G1 (Y )Q → G1 (R)Q = (R× )Q is injective.
Let Ye be the normalization of Y , and let Div(Ye ) denote the group of divisors
on Ye . Since the homomorphism div : R× → Div(Ye ), which sends f ∈ R× to theprincipal divisor div(f ),Thas a ﬁnite kernel and the group Div(Ye ) is a free abelianngroup, the intersection n (R× )` must be ﬁnite. Hence the homomorphism c1,1,Q
is injective. The claim is proved.
¤
3.3. Let us consider the exact sequence
∂

2
2
3
Hét
(X, Q` (2)) → Hét
(Xη , Q` (2)) −−ét→ HY,ét(X, Q` (2)).

Let Y sing denote the singular locus of Y equipped with the reduced scheme struc3
13ture. Let jét : HY,ét(X, Q` (2)) → Hét(Spec R, Q` (1)) denote the composite HY,ét(X, Q` (2)) →
3
sing
1
sing
1
HY \Y sing (X \ Y, Q` (2)) → Hét (Y \ Y, Q` (1)) → Hét (Spec R, Q` (1)), wherethe ﬁrst map is the pullback map, the second map is the isomorphism in [Gi, p.235,
Theorem 3.1], and the third map is the pullback map.
Lemma 3.3. For an element x ∈ K2 (Xη )Q , we have c2,2,Q (x) = 0 if and only ifjét ◦ ∂ét ◦ c2,2,Q (x) = 0.
2
2
(X ×S
Proof. By [SGA4-3, XII, Corollaire 5.5, p. 90], we have Hét
(X, Q` (2)) ∼= Hét2
2s, Q` (2)) ∼
H
(Y,
Q
(2)).
The
weightargument
shows
that
the
group
H
(Y,
Q
= ét
`
` (2))ét
is zero. Hence the homomorphism ∂ét is injective.
Let Z ⊂ Y be a closed subscheme of dimension zero. Consider the exact sequence3
3(X, Q` (2)) → HY3 \Z,ét (X \ Z, Q` (2)).
(X, Q` (2)) → HY,ét
HZ,étSince X is regular, the absolute purity theorem (cf. [Th, Corollary 1.10 (1.14),
p.123] and its proof. See [Fu, Introduction, p. 153] for a general statement and3(X, Q` (2)) is zero.
references.) implies that the group HZ,ét
1
The group Hét (Spec R, Q` (1)) is, again by purity, canonically isomorphic to the
inductive limit limZ HY3 \Z,ét (X \Z, Q` (2)) where Z runs over the closed subschemes−→of Y of dimension zero. It follows that jét is injective. This proves the claim.
¤
3.4. Proof of Proposition 3.1. Let K denote the ring of global sections of thesheaf of total quotient rings of OX . Let j 0 : K2 (Xη )Q → K2 (K)Q be the pullback

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

9homomorphism. We have the following commutative diagram∂Q

K2 (Xη )Q


j0y

−−−−→

G1 (Y )Qjy

K2 (K)Q
c2,2,K,Q y

−−−−→

∂K,Q

(R× )Qc1,1,Qy∂K,ét2
1
Hét(Spec K, Q` (2)) −−−−→ Hét(Spec R, Q` (1))

where the maps ∂K,Q and ∂K,ét are (the limit of) the maps in the localization
sequence of G-theory and étale cohomology respectively.
02
2
Let jét: Hét(Xη , Q` (2)) → Hét(Spec K, Q` (2)) be the pullback homomorphism.
Then the diagram
c2,2,Q

K2 (Xη )Q −−−−→
j0yc2,2,K,Q

2
Hét(Xη , Q` (2))

j 0
y ét

∂

−−−ét−→3
HY,ét(X, Q` (2))

j
y ét

∂K,ét2
1
K2 (K)Q −−−−−→ Hét(Spec K, Q` (2)) −−−−→ Hét(Spec R, Q` (1))

is commutative. From the two commutative diagrams above, we obtain the equalities
c1,1,Q ◦ j ◦ ∂Q = ∂K,ét ◦ c2,2,K,Q ◦ j 0 = jét ◦ ∂ét ◦ c2,2,Q
1
of homomorphisms from K2 (Xη )Q to Hét(Spec R, Q` (1)). Using Lemmas 3.2 and
3.3, the claim of Proposition 3.1 is proved.
¤
4. Elliptic sheaves and the determinant morphism
When Katz and Mazur compute the bad reduction of elliptic modular curves intheir book [Ka-Ma], they use extensively the Weil pairing ([Ka-Ma, p.87, (2.8)])
associated to an elliptic curve. Actually they use the morphism from the elliptic
modular curves to the spectrum of a ring of cyclotomic integers, which is inducedby the Weil pairing (see [Ka-Ma, p.281, 9.4.1]).
The aim of this section is to provide an analogue for Drinfeld modular curves.
For this, we do not give the actual pairing for Drinfeld modules, but construct what
we call the determinant morphism from the moduli of rank d Drinfeld modules tothe moduli of rank 1 Drinfeld modules (Section 4.4). The rough idea is to send a
Drinfeld module of rank d to its “determinant Drinfeld module”, that is, a maximal
exterior power. In the case of rank 2, this may be understood as the morphism
induced from the Weil pairing. For application, we only need the case of rank 2,
but since the exposition does not become simpler even if we restrict to the rank 2case, we give the construction in the case of general rank.
There are at least two ways of understanding the notion of the determinant of a
Drinfeld module. One is to use the theory of A-motives of Anderson and the other is
to use the theory of elliptic sheaves. The category of A-motives, which contains thecategory of Drinfeld modules, is equipped with a tensor structure. This direction
is pursued by van den Heiden [vdHe]. See his introduction for the details.
We use elliptic sheaves in this article. This notion is equivalent to that of Drinfeld
modules (see [Dr3], [Bl-St, Theorem 3.2.1, p. 149]). An elliptic sheaf is by deﬁnitiona vector bundle with some additional structure, and one expects the determinant

10

SATOSHI KONDO AND SEIDAI YASUDA

vector bundle to be equipped with the structure of an elliptic sheaf of rank 1. This isindeed the case (Lemma 4.3) and gives rise to the desired morphism. An advantage
of using elliptic sheaves is that they are equipped with certain canonical structure
at the inﬁnity prime. This is helpful when giving the adelic description (using theuniformization of the moduli space) of the determinant morphism.
In Section 4.1, we recall the basic properties of Drinfeld modular varieties. Via
the equivalence of the category of elliptic sheaves and the category of Drinfeld
modules, we regard them as the properties of the moduli of ellitpic sheaves. We
recall the deﬁnition of elliptic sheaves and its level structures in Section 4.2.1.
In section 4.3, we recall the uniformization of Drinfeld modular varieties. For
Drinfeld modular varieties, this has been given in the original paper by Drinfeld.
For the moduli of elliptic sheaves, the statement is given in [Bl-St, p.167, Theorem
4.3.5]. We give the details. In particular, the explicit construction of the morphism
αd (Section 4.3.3) is given. This will be needed in the proof of Proposition 4.5.
Section 4.4 is devoted to the construction of the determinant morphism. Using themoduli interpretation, we obatain the morphism only over C \ (Spec(A/I) ∪ {∞})
(we refer to the text for notation). Then the regularity of the moduli will give aunique extention to the morphism over C \ {∞} (Section 4.4.2).
The results of this section will (directly) be used only in Sections 6.1 and 9.3.
In Section 6.1, we will need that the ﬁeld of constants of the modular curve MI2 isthe coordinate ring of MI1 . The structure morphism is precisely given by the determinant morphism. In Section 9.4, we use the uniformization of modular varieties(Section 4.3.4, (4.2)).
We mention that the ﬁrst work in the actual construction of (the analogue of)the Weil pairing is by Hamahata [Ha] for the base ring A = Fq [t] (see below fornotation) and for Drinfeld modules of rank two using the theory of t-motives. Vanden Heiden [vdHe] also constructs the pairing for general A and for Drinfeld modules of general rank using the theory of A-motives. We have not pursued if theirconstructions give a morphism of moduli spaces.
4.1. Setting. Let C be a projective smooth geometrically irreducible curve overa ﬁnite ﬁeld Fq of q elements of characteristic p. For a closed point ℘ ∈ C, letκ(℘) denote the residue ﬁeld at ℘. We ﬁx a closed point ∞ ∈ C. Let A =H 0 (C \ {∞}, OC ) denote the coordinate ring of C \ {∞}.
We let k = Fq (C) denote the function ﬁeld of C. For a prime ℘ of k, we let k℘denote the completion of k at ℘. We regard a prime of k as a closed point of C
and vice versa. If ℘ 6= ∞ we denote by the same symbol ℘ the prime ideal of A
corresponding to ℘. Then the ℘-adic completion A℘ of A is equal to the ring ofintegers of k℘ . We let O∞ denote the ring of integers of k∞ .
Let d be a positive integer. Let I ⊂ A be a non-zero ideal. We let MId denote thefunctor which classiﬁes Drinfeld modules of rank d with a Drinfeld level I structure.
When Spec (A/I) contains two or more closed points, it is representable by an aﬃnescheme over Spec A ([Dr1, Proposition 5.3, p. 576]), and we will denote the latterby the same symbol. We will recall the deﬁnition of a Drinfeld module and thatof a Drinfeld level I structure in Section 5.1. The scheme MId has the following
properties.
Lemma 4.1. Suppose that Spec (A/I) contains two or more closed points.
(1) The scheme MId is a regular scheme of pure dimension d.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

11(2) The structure morphism MId → Spec A is of ﬁnite type, ﬂat, and surjective. It is smooth when restricted to the open subscheme UI = Spec A \Spec (A/I) ⊂ Spec A.
(3) For an ideal J ⊂ I, the canonical “level-lowering” morphism MJd → MId is
ﬁnite ﬂat.
Proof. All assertions, except the ﬂatness and the surjectivity in (2), are immediate
consequences of [Dr1, Proposition 5.3, p 576] and its corollary. The ﬂatness follows
from the local description of MId in [Dr1, Proposition 5.4, p. 576]. The surjectivity
follows from [Dr1, §8, Corollary, p 585].
¤
1=
We recall the fundamental properties of the moduli scheme MI1 . Let MI/k

MI1 ×Spec A Spec k.
Lemma 4.2. Suppose that Spec (A/I) contains two or more closed points. The
1
scheme MI/kover Spec k is isomorphic to the spectrum of a ﬁnite abelian extensionof k which is completely split at ∞ and unramiﬁed outside the primes dividing I.
1
The scheme MI1 over Spec A is isomorphic to the integral closure of Spec A in MI/k.
Proof. This follows from [Dr1, §8, Theorem 1, p. 585] and its proof.

¤

4.2. Elliptic sheaves.
4.2.1. We recall the deﬁnition of an elliptic sheaf. Let S be a scheme over C. Let dbe a positive integer. An elliptic sheaf of rank d over S is a sequence (Ei , ji , ti )i∈Z ,where Ei are locally free OC×Spec Fq S -modules of rank d and where ji : Ei → Ei+1 ,ti : τ Ei → Ei+1 are injective OC×Spec Fq S -linear homomorphisms. Here we putτ
Ei := (idC × FrobS )∗ Ei , where FrobS denotes the endomorphism on S which is
the identity on the underlying topological space of S and which is the q-th power
endomorphism on the structure sheaf of S. The following conditions should hold.
(1) For each i the following diagram is commutative:Eixti−1 

ji−−−−→ Ei+1
x

ti 
τ

τ

ji−1

Ei−1 −−−−→ τ Ei .

(2) For each i there exists an isomorphism
Ei+d·deg(∞) ∼= Ei (∞) := Ei ⊗O

C×Spec F S
q(OC (∞) £ OS )where OC (∞) £ OS = pr∗1 (OC (∞)) ⊗ pr∗2 (OS ) is an invertible sheaf on
C ×Spec Fq S such that the composite
Ei ,→ Ei+1 ,→ · · · ,→ Ei+d·deg(∞) ∼= Ei (∞)
is the canonical embedding.
(3) The direct image of Ei+1 /ji (Ei ) under the projection to the second factor
prS : C ×Spec Fq S → S is a locally free OS -module of rank one.
(4) For each i ∈ Z, the cokernel Coker ti is supported on the graph of a morphism ιi : S → C over Spec Fq , and is the direct image of a locally free(ιi ,idS )module on S of rank one by the morphism S −−−−−→ C ×Spec Fq S. Moreover, when i = −d(g − 1), the morphism ι−d(g−1) equals the structure
morphism S → C, where g is the genus of the curve C.

12

SATOSHI KONDO AND SEIDAI YASUDA(5) For any closed point s ∈ S, one has deg(E0 |C×Spec Fq {s} ) = d(g − 1) + 1.
4.2.2. Level structure. Let I ⊂ A be a nonzero ideal. Let S be a scheme over C \Spec (A/I). Let (Ei , ji , ti )i∈Z be an elliptic sheaf of rank d over S. The morphisms
ji for i ∈ Z identify the restrictions Ei |Spec (A/I)×Spec Fq S . The homomorphism t0
induces∼tI : τ E0 |Spec (A/I)×
S → E1 |Spec (A/I)×
S = E0 |Spec (A/I)×S.
Spec Fq

Spec Fq

Spec Fq

A level I structure on (Ei , ji , ti )i∈Z is an isomorphism∼=⊕dι : E0 |Spec (A/I)×Spec Fq S −→ OSpec(A/I)×Spec F S
qsuch that the composite
τ

∼
=

⊕d
ι : τ E0 |Spec (A/I)×Spec Fq S −→ τ (OSpec (A/I)×Spec Fq S )⊕d = OSpec(A/I)×Spec F S
qequals ι ◦ tI .
If S is a scheme over Spec A, there is a canonical one-to-one correspondence
between the isomorphism classes of Drinfeld modules of rank d over S with a Drinfeld level I structure and those of elliptic sheaves of rank d over S with a level I
structure ([Dr3], [Bl-St, Theorem 3.2.1, p. 149]).
d
Let MI,C\Spec(A/I) denote the moduli stack over C \ Spec (A/I) of the elliptic(d)sheaves of rank d with a level I structure, introduced and denoted by E``X,I in[Bl-St, Deﬁnition 3.1.11, p. 148]. It is an algebraic stack in the sense of DeligneMumford. Using the one-to-one correspondence mentioned in the previous paradgraph, we identify the base change MI,C\Spec(A/I) ×C\Spec (A/I) (Spec A\Spec (A/I))dwith the stack MI ×Spec A (Spec A \ Spec (A/I)).
4.3. Uniformization of Drinfeld modular varieties. We recall in this section
the uniformization of Drinfeld modular varieties at inﬁnity. We give the adelic
description using elliptic sheaves. Many of the statements are found in the paper[Bl-St, §4] by Blum and Stuhler; we give the explicit details for later use.
We denote by A the k-algebra of adeles of k, by A∞ the k-algebra of ﬁnite adelesb the projective limit lim A/I where I runs over all non-zero idealsof k, and by A←−I
b is a subring of A∞ and A∞ = Ab ⊗A k. Let d be a positiveof A. We recall that A
integer. Let k∞ denote the completion of k at ∞, and O∞ be the ring of integers.
We let π∞ ∈ O∞ denote a uniformizer.
4.3.1. Elliptic sheaves at inﬁnity. We follow [Bl-St, p.163] and deﬁne a functor
(d)
c
Ell
from Nilp to the category of sets as follows. Here Nilp denotes the categoryof commutative O∞ -algebras R such that the image of π∞ in R is nilpotent.
For an object R in Nilp, we denote by FrobR the q-power endomorphism of R.
b Fq N denote
For a ﬁnitely generated O∞ -module M and an Fq -algebra N , we let M ⊗the (π∞ ⊗ 1)-adic completion of M ⊗Fq N . For a ﬁnite dimensional k∞ -vector spaceb Fq N 0 denote the inductive limit lim Λ⊗b Fq N 0
V and an Fq -algebra N 0 , we let V ⊗−→Λwhere Λ runs over the O∞ -lattices of V .
(d)c (R) is the set of isomorphism classes of
Let R be an object of Nilp. Then Ellsequences (Λi , ti )i∈Z whose descriptions are given below:⊕d bb Fq R-submodule of k∞• For each i ∈ Z, the Λi is an O∞ ⊗⊗Fq R. The Λi ’s
satisfyON THE RATIONAL K2 OF AN ELLIPTIC CURVE

13(1) · · · ⊂ Λi−1 ⊂ Λi ⊂ Λi+1 ⊂ · · · , and−1(2) Λi+d·deg(∞) = (π∞⊗ 1)Λi .
b Fq R-module, locally free of rank d.
• Each Λi is, as an O∞ ⊗• For each i, there exists a morphism αi : Spec R → Spec κ(∞) such that theαiquotient Λi+1 /Λi is supported on the graph of the composite Spec R −→Spec κ(∞) → Spec O∞ where the second morphism is the canonical closed
immersion.
• For each i, the quotient Λi /Λi+1 , regarded as an R-module via the canonicalb Fq R which sends r ∈ R to 1 ⊗ r, is locally freehomomorphism R → O∞ ⊗of rank one.
b Fq R• For each i, the ti is an injective homomorphism τ Λi → Λi+1 of O∞ ⊗τb Fq R-module Λ, we denote by Λ the base change
modules. Here for an O∞ ⊗b
Λ ⊗O∞ ⊗O⊗
∞ Fq R.
b F R, id⊗Frobb
Rq• For each i, the diagram⊂Λi+1 −−−−→
x

ti 
τ

⊕d b
k∞
⊗Fq R
°
°
°

⊂
⊕d b
⊗Fq R)
Λi −−−−→ τ (k∞is commutative.
• For each i, the cokernel of ti is supported on the graph of a morphism
Spec R → Spec O∞ over Spec Fq . Moreover, when i = −d(g − 1), thishomomorphism equals the structure morphism Spec R → Spec O∞ . Here gdenotes the genus of C.
• Each Coker ti , regarded as an R-module via the canonical homomorphismb Fq R which sends r ∈ R to 1 ⊗ r, is locally free of rank one.
R → O∞ ⊗4.3.2. Let I ⊂ A be a non-zero ideal. Let KI denote the compact open subgroupb → GLd (A/I)] of GLd (A∞ ). For a formal scheme F over Spf O∞ on
Ker[GLd (A)which the group GLd (k) acts from the left, we deﬁne the formal stack
GLd (k)\(GLd (A∞ )/KI × F )³`´
as the quotient of the product
GLd (A∞ )/KI Spf O∞ ×Spf O∞ F by the diagonal`
action of GLd (k). Here GLd (k) acts on GLd (A∞ )/KI Spf O∞ from the left on theindex set.
If we regard F as a covariant functor from Nilp to the category of sets, then theformal stack GLd (k)\(GLd (A∞ )/KI × F ) is the category ﬁbered in groupoids overthe category Nilpop whose description is given as follows:• An object in GLd (k)\(GLd (A∞ )/KI × F ) is a triple (R, Q, α) which consists of an object R in Nilp, a left GLd (k)-torsor Q over Spec R, and a
GLd (k)-equivariant morphism α : Q → GLd (A∞ )/KI ×(F ×Spf O∞ Spec R).
Here F ×Spf O∞ Spec R denote the product of F and the covariant functor represented by R from Nilp to the category of sets, which we regard
as a scheme over Spec R, and GLd (A∞ )/KI × (F ×Spf O∞ Spec R) denotes
the disjoint union of copies of the scheme F ×Spf O∞ Spec R indexed by
the set GLd (A∞ )/KI where the group GLd (k) acts both on the index set
GLd (A∞ )/KI and the scheme F ×Spf O∞ Spec R.

14

SATOSHI KONDO AND SEIDAI YASUDA• For two objects (R, Q, α), (R0 , Q0 , α0 ) in GLd (k)\(GLd (A∞ )/KI × F ), theset of morphisms from (R, Q, α) to (R0 , Q0 , α0 ) is the set of pairs (f, φ) where∼=f : R0 → R is a morphism in Nilp and φ : Q0 ×Spec R0 SpecR −→ Q is an
isomorphism of GLd (k)-torsors over Spec R such that the diagram
α0 ×idQ0 ×Spec R0 SpecR −−−−→ GLd (A∞ )/KI × (F ×Spf O∞ Spec R)

°

°
φy
°
Q

α

−−−−→ GLd (A∞ )/KI × (F ×Spf O∞ Spec R)

is commutative.
• The functor GLd (k)\(GLd (A∞ )/KI × F ) → Nilpop is given by sending(R, Q, α) to R.
For an object R in Nilp, we denote by (GLd (k)\(GLd (A∞ )/KI × F ))(R) the ﬁberof GLd (k)\(GLd (A∞ )/KI × F ) over R. We will use the notation introduced in this(d)cb (d) , where the formal scheme Ωb (d)paragraph only for F = Elland for F = Ωwill appear in Section 4.3.4. In both cases, F admits a left action of the group
GLd (k∞ ), and hence of GLd (k).
cd denote the formal completion of the algebraic stack M d
4.3.3. Let M
I,∞
I,C\Spec (A/I)
dalong its special ﬁber MI,∞at ∞ ∈ C (see [Bl-St, Theorem 4.3.5, p. 167]). We(d)c ) of formal stacks.
cd → GLd (k)\(GLd (A∞ )/KI × Elldeﬁne a morphism αd : M
I,∞
We use the following notation. For a projective A-module P of ﬁnite rank, wedenote by Pe the locally free OSpec A -module of ﬁnite rank corresponding to P . For
two schemes X, S over Spec Fq , we denote by X × S the ﬁber product X ×Spec Fq S.
For an OX -module F, we denote by F £Fq OS the pullback of F under the ﬁrst
projection X × S → X.
Let R be an object in Nilp. We construct a map of R-valued points. Let
E = (Ei , ji , ti )i∈Z be an elliptic sheaf over S = Spec R with a level I structure.
The restriction Ei |(Spec A)×S does not depend on i in the sense that the injection
ji : Ei ,→ Ei+1 induces an isomorphism Ei |(Spec A)×S ∼= Ei+1 |(Spec A)×S . By [St,§3, Proposition1, p. 508], there exists a projective A-module P = PE of rank dsuch that the contravariant functor Q0E from the category of schemes over S to thecategory of sets which associates, to a scheme T over S, the set of isomorphisms
ψ : Ei+1 |(Spec A)×T ∼= Pe £Fq OTof O(Spec A)×T -modules such that the diagramψ

Ei+1 |(Spec A)×T −−−−→x
ti |(Spec A)×T 

Pe £Fq OT
°
°
°

τ

ψ
(τ Ei )|(Spec A)×T −−−−→ τ (Pe £Fq OT )is commutative, is representable by an étale left AutA (P )-torsor over S. Here thesymbol |(Spec A)×T denotes the pullback to (Spec A) × T .
∼=→ k ⊕d . We regard the group AutA (P )We ﬁx an isomorphism µ = µE : P ⊗A k −as a subgroup of GLd (k) via this isomorphism. We deﬁne the left GLd (k)-torsor
QE = QE,µ over S as QE = GLd (k) ×AutA (P ) Q0E .

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

15

Let T = Spec R0 be a connected aﬃne scheme over S and let ψ ∈ Q0E (T ). (Notethat R0 is then an object in Nilp.) For each i ∈ Z, let us consider the completionof the pullback of Ei to C × T along Spec κ(∞) × T . Via the isomorphism ψ,b Fq R0 -lattice Λ0ψ,i in (P ⊗A k∞ )⊗b Fq R0 . We denote bythis completion gives an O∞ ⊗b : (P ⊗A k∞ )⊗b Fq R0 =Λψ,i the image of Λ0ψ,i under the isomorphism (µ ⊗ id)⊗id
⊕d
0
⊕d
0
0
∼
b Fq R = (k ⊗k k∞ )⊗b Fq R = k∞ ⊗b Fq R . The homomorphisms((P ⊗A k) ⊗k k∞ )⊗ji : Ei → Ei+1 and ti : τ Ei → Ei+1 then induce homomorphisms jψ,i : Λψ,i → Λψ,i+1(d)c (R0 ).
and tψ,i : τ Λψ,i → Λψ,i+1 . These data deﬁne an element (Λψ,i , tψ,i )i∈Z in Ell
Suppose that the level I structure on the elliptic sheaf (Ei , ti , ji )i∈Z is given. Via theisomorphism ψ, this level structure gives an isomorphism Pe £Fq OT |Spec (A/I)×T ∼
=
⊕d
OSpec
.
Since
T
is
connected,
to
give
such
an
isomorphismisequivalentto(A/I)×T
giving an isomorphism ιψ : P/IP ∼= (A/I)⊕d of A-modules.
∼=
For each prime ℘ 6= ∞ of k, take an isomorphism eιψ,℘ : A⊕d→ P ⊗A A℘ such
℘ −that the diagram
A⊕d
℘


y

e
ιψ,℘

−−−−→

P ⊗A A℘


y

ιψ,℘

(A℘ /IA℘ )⊕d −−−−→ (P/IP ) ⊗A A℘
is commutative, where the vertical homomorphisms are the natural projections andthe isomorphism ιψ,℘ is the one induced by the isomorphism ιψ . Let gψ,℘ be thed × d matrix with entries in k℘ such that the compositeeιψ,℘ ⊗idµ⊗id−−−−→ P ⊗A A℘ ⊗A℘ k℘ = P ⊗A k ⊗k k℘ −−−→ k ⊕d ⊗k k℘ = k℘⊕d
k℘⊕d = A⊕d℘ ⊗A℘ k℘ −is supplied by the right multiplication by gψ,℘ (here we regard k℘⊕d as the space ofrow vectors). Then class of the element (gψ,℘ )℘6=∞ ∈ GLd (A∞ ) in GLd (A∞ )/KI
does not depend on the choice of eιψ,℘ .
We deﬁne the AutA (P )-equivariant morphism αE0 : Q0E → GLd (A∞ )/KI ×
(d)
c
(Ell
× Spec R) as the morphism which send ψ ∈ Q0E (T ) for each connectedaﬃne scheme T over S to ((gψ,℘ )℘6=∞ , (Λψ,i , tψ,i )i∈Z ). The morphism αE0 induces(d)ca GLd (k)-equivariant morphism αE : QE → GLd (A∞ )/KI × (Ell× Spec R)
which does not depend on the choice in the sense that for two isomorphisms∼∼==µ1 , µ2 : P ⊗A k −→ k ⊕d , there exists a canonical isomorphism ν : QE,µ1 −→ QE,µ2 ofleft GLd (k)-torsors such that the diagram(d)× Spec R)(d)× Spec R)αc
QE,µ1 −−−E−→ GLd (A∞ )/KI × (Ell

°

°
νy
°
α
c
QE,µ2 −−−E−→ GLd (A∞ )/KI × (Ellis commutative. We deﬁne the morphism αd as the morphism induced by thefunctor(d)dc )(R)cI,∞
M(R) → (GLd (k)\(GLd (A∞ )/KI × Ell
which sends (Ei , ti , ji )i∈Z to (QE , αE ) for each object R in Nilp.
Using the argument in the proof of [St, §3, Theorem 2, p. 514] (where the levelstructure is not considered), one can prove that αd is an isomorphism.

16

SATOSHI KONDO AND SEIDAI YASUDA

4.3.4. Let Xd be the Drinfeld symmetric space (see [Bl-St, 4.2, p. 162] where it(d)is denoted by Ωrig ) for GLd . It is a rigid analytic space over k∞ , and the group
GLd (k∞ ) acts from the left on Xd . When d = 1, the space X1 is just one point.
b (d) over O∞ , which is
The rigid analytic space Xd has a canonical formal model Ωconstructed by Deligne and whose deﬁnition is given in [Gen, III, Déﬁnition 1.2.2,b d ).
p. 73] (where it is denoted by Ω(d)c and the functor GO of Genestier ([Gen, III,One can verify that the functor Ell
Déﬁnition I.4.3.3, p. 29]) (where their O is our O∞ ) are isomorphic by comparing(d)cour deﬁnition of Ell
in Section 4.3.1 and the deﬁnition by Genestier [Gen, Variant 4.4.2 de la déﬁnition 4.3.3, p. 30]. It is proved in [Gen, 3.1, Théorème 1.1, p. 94]
b (d) and GO are isomorphic. Hence we obtain a uniformizationthat the functors Ω
cd ∼b (d) )
(4.1)
M
= GLd (k)\(GLd (A∞ )/KI × Ω
I,∞where the isomorphism is given by the composition of the isomorphism αd and the(d)c (Recall the convention in Section 4.3.2). Hereb (d) and Ell
isomorphism between Ωb (d) denotes the disjoint union of the copies of Ωb (d) indexed by theGLd (A∞ )/KI × Ω∞
set GLd (A )/KI , on which the group GLd (k) acts from the left via its action onb (d) .
GLd (A∞ )/KI and on Ω
For a quasi-projective scheme X over Spec k∞ , let X an denote the rigid analytic
space associated to X. By taking the generic ﬁber in the sense of Raynaud of bothsides of (4.1), one obtains the uniformization of analytic Drinfeld modular varieties(cf. [Dr1], [Bl-St, Remark 4.3.1 ii), p. 166, Theorem 4.3.5, p. 167]):(4.2)(M d )an ∼= GLd (k)\(GLd (A∞ )/KI × Xd ).
I/k∞4.4. The determinant morphism. In this section, we construct what we call adeterminant morphism.
4.4.1. Let I $ A be a non-zero ideal. Let S be a scheme over C \ Spec (A/I).
Given an elliptic sheaf (Ei , ti , ji )i∈Z over S of rank d with a level I structure ι :⊕d
E0 |Spec (A/I)×Spec Fq S ∼= OSpec(A/I)×Spec F S , we deﬁne its determinant elliptic sheafq

to be the triple (Fi , t00i , ji00 ), where Fi is the exterior power Fi = ∧d Ei−(d−1)(g−1)
which is an invertible OC×Spec Fq S -module, and t00i and ji00 are the homomorphisms
t00i = ∧d ti−(d−1)(g−1) : τ Fi ∼= ∧dτ Ei−(d−1)(g−1) → Fi+1 , and ji00 = ∧d ji−(d−1)(g−1) :Fi → Fi+1 , respectively.
Lemma 4.3. The triple (Fi , t00i , ji00 )i∈Z is an elliptic sheaf of rank one.

Proof. We need to check Conditions (1)-(5) in Section 4.2.1 for (Fi , , t00i , ji00 )i∈Z .
Condition (1) follows from the functoriality of exterior power operations. Conditions (3) and (4) follow from Lemma 4.4 below. Condition (5) follows since thehighest exterior power operation does not change the degree of locally free sheaves.
Hence it remains to check Condition (2).
Let us give a proof that Fi+deg(∞) ∼= Fi (∞). We may assume that S is connected
and is of ﬁnite type over Spec A. By Conditions (2) and (3) in the deﬁnition of
elliptic sheaves, the scheme {∞} × S decomposes into a disjoint union {∞} ×`deg(∞)
S = j=1 Sj of deg(∞) connected components, and for each i the OC×Spec Fq S module Fi+1 /ji00 (Fi ) is the direct image of an invertible module on Sji for someji . By Conditions (3) and (4) in the deﬁnition of elliptic sheaves, the map ti

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

17

induces an isomorphism τ (Fi /Fi−1 ) ∼= Fi+1 /Fi for each i. This implies that thecomponents Sji , · · · , Sji+deg(∞)−1 are pairwise distinct. Hence Fi+deg(∞) /Fi is the
direct image of an invertible module on {∞}×S. Since Fi+deg(∞) /Fi is a submoduleof Fi+d·deg(∞) /Fi ∼¤= Fi (d∞)/Fi , it equals Fi (−∞)/Fi .
Lemma 4.4. If f : E → E 0 is a homomorphism of locally free sheaves of rank d on ascheme T such that Coker f is isomorphic to the direct image of an invertible sheaf
on a closed subscheme T 0 ⊂ T , then the cokernel of the homomorphism ∧d E → ∧d E 0
is also isomorphic to the direct image of an invertible sheaf on T 0 .
Proof. It suﬃces to prove the case where T = Spec R is the spectrum of a commutative local ring R, and T 0 is the closed point of T . Let f : R⊕d → R⊕d be an
R-linear endomorphism which is expressed by a d × d matrix F . Suppose that thecokernel Coker f of f is generated as an R-module by at most one element. The
claim is that Coker f is isomorphic to R/(det F ).
Let e ∈ R⊕d be an element in the preimage of a generator of Coker f under thequotient map R⊕d → R⊕d /f (R⊕d ) = Coker f . Let e1 , . . . , ed be the standard basisof R⊕d .
It follows from Nakayama’s lemma that there exists an integer i ∈ {1, . . . , d}such that R⊕d is generated by e and {f (ej ) | j 6= i}. Let h : R⊕d → R⊕d be
the endomorphism which sends ei to e and sends ej to f (ej ) for j 6= i. Then h issurjective.
Let m ⊂ R denote the maximal ideal. Because h is an isomorphism modulo m,
its determinant is non-zero modulo m which means that det(h) is invertible. Thisimplies, using Cramer’s rule, that h is an isomorphism.
Hence {e}P
∪ {f (ej ) | j 6= i} forms a basis of R⊕d . Write f (ei ) in the formf (ei ) = be + j6=i bj f (ej ) with b, bj ∈ R. Then we see that Coker f ∼= R/(b). OnP−1
the other hand the composite h ◦ f sends ei to bei + j6=i bj ej and sends ej to
ej for j 6= i. Hence the determinant of the d × d matrix corresponding to h−1 ◦ f
is equal to b. This proves the claim.
¤
4.4.2. Using the isomorphism which sends the element (1, 0, . . . , 0)∧· · ·∧(0, . . . , 0, 1)⊕dd ⊕dof ∧d OSpec(A/I)×Spec F S to 1 ∈ OSpec (A/I)×Spec Fq S , we identify ∧ OSpec (A/I)×Spec F S
qwith OSpec (A/I)×Spec Fq S . Since F0 = ∧d E0 , the level I structureq⊕dι : E0 |Spec (A/I)×Spec Fq S ∼= OSpec(A/I)×Spec F S
qon (Ei , ti , ji )i∈Z induces a level I structure∧d ι : F0 |Spec (A/I)×Spec Fq S ∼= OSpec (A/I)×Spec Fq S
on (Fi , t00i , ji00 )i∈Z .
The correspondence, which takes an elliptic sheaf to its determinant elliptic sheaf,
gives rise to a morphismd1
wC\Spec (A/I) : MI,C\Spec(A/I) → MI,C\Spec (A/I)

over C \ Spec (A/I). We set UI = Spec A \ Spec (A/I). The morphism wC\Spec (A/I)
induces the morphism MId ×Spec A UI → MI1 ×Spec A UI . Suppose that Spec (A/I)
contains two or more closed points. From Lemma 4.2, we know that the scheme
MI1 is isomorphic to the integral closure of Spec A in MI1 ×Spec A UI . Since MId
is regular (in particular normal), the morphism MId ×Spec A UI → MI1 ×Spec A UI18

SATOSHI KONDO AND SEIDAI YASUDA

extends uniquely to a morphism w : MId → MI1 by [EGAII, (6.3.9), p. 119]. We
call the morphism w the determinant morphism.
4.5. Adelic description of the determinant morphism. Let d be a positive
integer. Since X1 is a point, there exists a canonical morphism(4.3)

r : GLd (k)\(GLd (A∞ )/Kd,I × Xd ) → GLd (k)\(GLd (A∞ )/Kd,I × X1 )r0−→ k × \((A∞ )× /K1,I × X1 )where the last map r0 is the one induced by the determinant homomorphism det :GLd (A∞ ) → (A∞ )× .
Proposition 4.5. The morphism r0 in (4.3) is bijective. The morphism r in (4.3)
is compatible with the determinant morphism w : MId → MI1 , that is, the diagram
∼
=

(4.4)

d
)an −−−−→ GLd (k)\(GLd (A∞ )/Kd,I × Xd )
(MI/k
∞



wy
ry
∼
=

1
(MI/k
)an −−−−→∞

k × \((A∞ )× /K1,I × X1 )is commutative. Here, by abuse of notation, we let w denote the analytiﬁcation of
the base change to k∞ of the determinant morphism w. The horizontal isomorphisms are those of (4.2).
Proof. The surjectivity of r0 is clear. To show the injectivity, let X1 = {x} andsuppose that r0 (GLd (k)g1 Kd,I , x) = r0 (GLd (k)g2 Kd,I , x). Replacing g2 by anotherelement in GLd (k)g2 Kd,I , we may assume that det(g1 ) = det(g2 ). Since SLd (k) isdense in SLd (A∞ ) by the strong approximation theorem, the intersection SLd (k) ∩g1 Kd,I g2−1 is non-empty. This implies that GLd (k)g1 Kd,I = GLd (k)g2 Kd,I . Hencer0 is injective.
cd → Mc1 be the morphism of formal stacks over Spf O∞ induced
Let w
b∞ : MI,∞
I,∞
by the morphism wC\Spec (A/I) . We will show that the following diagram∼=
cd −−−
b (d) )M−→ GLd (k)\(GLd (A∞ )/Kd,I × Ω
I,∞




w
b∞ y
r
by
∼
=
c1 −−−
M−→I,∞b (1) ).
k × \((A∞ )× /K1,I × Ω

where the horizontal maps are the isomorphisms given in (4.1) and the right vertical
map rb is the one induced by det : GLd (A∞ ) → (A∞ )× and the structure morphismb (d) → Spf O∞ = Ωb (1) , is commutative. Then by taking the generic ﬁber in theΩsense of Raynaud, we obtain the commutativity of the diagram (4.4). Since theisomorphism (4.1) is obtained as the composite
(d)

αd
d
c
cI,∞
M
−−→
GLd (k)\(GLd (A∞ )/KI × Ellb (d) ),)∼= GLd (k)\(GLd (A∞ )/KI × Ω(d)∼cwhere the second isomorphism is the one induced by the isomorphism Ell=(d)bΩ which we have explained in Section 4.3.4, the claim follows from Lemma 4.6
below.
¤

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

19

Lemma 4.6. The following diagram of formal stacks is commutative:
(d)

αd
c
cd −−−M−→ GLd (k)\(GLd (A∞ )/Kd,I × Ell
I,∞
∼
=




w
b∞ y
r
by
α1
c1 −−−
M
−→
I,∞
∼
=

(1)

c
k × \((A∞ )× /K1,I × Ell)).

Here the right vertical map is the one induced by det : GLd (A∞ ) → (A∞ )× .
Proof. Let R be an object in Nilp. It suﬃces to prove that the diagram(d)c
cd (R) −−−−→ GLd (k)\(GLd (A∞ )/Kd,I × Ell
M
I,∞




w
b∞ y
r
by
c1 (R) −−−−→
M
I,∞

(1)

c
k × \((A∞ )× /K1,I × Ell

(R))

(R)).

of groupoids is commutative up to functorial isomorphisms. Let y be an object
cd (R), that is, y is an elliptic sheaf E = (Ei , ji , ti ) over Spec R with a levelin M
I,∞
I structure. Let z = w
b∞ (y). Then, by deﬁnition, z is an elliptic sheaf F =(Fi , ji00 , t00i ) over Spec R with a level I structure, where Fi = ∧d Ei−(d−1)(g−1) , t00i =∧d ti−(d−1)(g−1) and ji00 = ∧d ji−(d−1)(g−1) .
For a commutative ring L, let σL : ∧d (L⊕d ) ∼= L denote the L-linear isomorphism
which sends the element (1, 0, . . . , 0)∧· · ·∧(0, . . . , 0, 1) of ∧d (L⊕d ) to 1 ∈ L. Let PE ,Q0E , QE , and µE be as in Section 4.3.3. We can take PF and µF so that PF = ∧d PE
and µF = σk ◦ ∧d µE . Let T = Spec R0 be a connected aﬃne scheme over Spec R
such that the AutA (PE )-set Q0E (T ) is non-empty. Take an element ψE in Q0E (T ).
Then the isomorphism ψF := ∧d ψE is an element in Q0F (T ).
Let ΛψE ,i for each i ∈ Z, and eιψE ,℘ , gψE ,℘ for each prime ℘ 6= ∞ of k be as in−1
Section 4.3.3. Then we can take eιψF ,℘ so that eιψF ,℘ = ∧deιψE ,℘ ◦ σA
. Let ΛψF ,i for℘deach i ∈ Z be as in Section 4.3.3. Then we have ΛψF ,i = σk∞ ⊗b F R0 (∧ ΛψE ,i ). Byq
the deﬁnition of the matrix gψE ,℘ , the composite∧d eιψ ,℘ ⊗idEd∧d k℘⊕d = (∧d A⊕d℘ ) ⊗A℘ k℘ −−−−−−−→ (∧ (P ⊗A A℘ )) ⊗A℘ k℘∧d µE ⊗id

= (∧d (P ⊗A k)) ⊗k k℘ −−−−−−→ (∧d k ⊕d ) ⊗k k℘ = ∧d k℘⊕d
is supplied by the multiplication by det gψE ,℘ . Hence we see from the description of the isomorphism αd given in Section 4.3.3 that the image of z under(1)∼=c )(R) is equal to the k × c1 (R) −→ (k × \((A∞ )× /K1,I × Ell
the equivalence M
I,∞

(1)

c
equivariant morphism k × ×AutA (PF ) Q0F → k × \((A∞ )× /K1,I × (Ell×Spf O∞Spec(R)) which is induced by the Aut(PF )-equivariant morphism α10 (z) : Q0F →(1)
c ×Spf O Spec(R)) characterized by the following propk × \((A∞ )× /K1,I × (Ell∞erty: for each connected aﬃne scheme T = Spec R0 over Spec R and for each element ψE in Q0E (T ), the image α10 (z)(ψF ) of ψF ∈ Q0F (T ) under α10 (z) is class of(1)
c (R0 ). This proves the((det gψ ,℘ )℘6=∞ , (Λψ ,i , σ b 0 ◦∧d ti )i∈Z ) ∈ (A∞ )× × Ell
Eclaim.

F

k∞ ⊗Fq R¤

20

SATOSHI KONDO AND SEIDAI YASUDA

5. Deuring’s theory for Drinfeld modules of rank two
The main goal of this section is to construct an isomorphism (5.4), and to prove
Proposition 5.4. They are the only results to be used in other sections.
Deuring’s theory for elliptic curves applies to the Drinfeld modular context. (Werefer to the introduction in [Gek3] for the historical account including the case ofDrinfeld modules.) This gives a description of the category of supersingular Drinfeld
modules of rank two in terms of the category of left O-modules, where O is theendomorphism ring of a supersingular Drinfeld module.
As a consequence of the equivalence of categories (Proposition 5.1), we obtain
the isomorphism (5.4), describing the set of supersingular points as a double coset
space. This gives rise to an action of the non-(℘, ∞)-part GL2 (A℘,∞ ) of the adeles.
On the other hand, there is a natural action of GL2 (A∞ ) coming from the action
on the moduli spaces as described by Drinfeld. We show in Proposition 5.4 thatthe two actions on the set of supersingular points are compatible.
The story is classical but we were not able to ﬁnd a written account of thismaterial. We think it is necessary to give the details when one is to check thecompatibility of the actions (Proposition 5.4).
In Section 5.1, we recall the deﬁnition of Drinfeld modules for completeness.
Section 5.2.1 is devoted to Deuring’s theory (Proposition 5.1). We note that thecorrespondence between the isomorphism classes (i.e., the objects of the categories)
are usually found in the literature; we need the correspondence between the morphisms and of some isogenies. In Section 5.2.2, we consider the case with levelstructures. Section 5.2.3 contains the main statements (the isomorphism (5.4) and
Proposition 5.4) to be used in Lemmas 7.1 and 7.2.
5.1. Drinfeld modules.
5.1.1. We recall the deﬁnition of a Drinfeld module. Let S be a scheme over Spec A.
Let d be a positive integer. A Drinfeld module of rank d over S is a scheme E of
A-modules over S satisfying the following three conditions.
(1) Zariski locally on S, the scheme E is isomorphic to additive group scheme
Ga as a commutative group scheme.
(2) If we denote the A-action on E by ϕ : A → EndS-group (E), then, for everya ∈ A\{0}, the a-action ϕ(a) : E → E on E is ﬁnite, locally free of constantdegree |a|d∞ .
(3) The A-action on Lie E induced by ϕ coincides with the A-action on Lie E
which comes from the structure homomorphism A → Γ(S, OS ).
5.1.2. Drinfeld level structure and modular varieties. Let d be a positive integer.
Let E be a Drinfeld module of rank d over a scheme S over Spec A. Let E(S)be the A-module of the sections of E → S. We regard an element in E(S) as aneﬀective Cartier divisor in E/S (in the sense of [Ka-Ma, 1.1.1, p. 3]). Let I ⊂ A
be a non-zero ideal. Let E[I] denote the I-torsion part of E. If we take generators
a1 , . . . , am ∈ I of I, then E[I] is equal to the ﬁber product of the zero section
S → E m (here E m denotes the m-fold ﬁber product of E over S) and the morphism
E → E m which is the composition of the morphism (a1 , . . . , am ) : E m → E m
with the diagonal embedding E → E m . A Drinfeld level I structure
on E is aPhomomorphism φ : (I −1 /A)⊕d → E(S) of A-modules such that a∈(I −1 /A)⊕d φ(a)equals E[I] as an eﬀective Cartier divisor in E/S.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

21

5.2. Supersingular Drinfeld modules of rank two. A Drinfeld module ofrank two over Spec κ(℘) is called ordinary (resp. supersingular) if the corresponding formal A℘ -module is of height one (resp. two) (see [Dr1, §1, p. 563] for thedeﬁnition of formal A℘ -modules). We refer to [Gek3, Proposition 4.1, p. 195] forother equivalent deﬁnitions.
5.2.1. Adelic description. A supersingular Drinfeld module E of rank two overSpec κ(℘) is known to exist. We ﬁx one such E and denote it by E0 . Put
O = EndA-mod/κ(℘) (E0 ). Here EndA-mod/κ(℘) denotes the set of endomorphismsof schemes of A-modules over Spec κ(℘). Then O is an A-algebra. Moreover,
B = O ⊗A k is a quaternion algebra over k which ramiﬁes exactly at ℘ and at ∞,and O is a maximal A-order of B (this is due to Drinfeld ([Dr2, §2, p. 161-163])).
We say that a left O-module M is invertible if M ⊗A A℘0 is a free O ⊗A A℘0 -moduleof rank one for each prime ℘0 6= ∞. Let M1 and M2 be two invertible left Omodules. We say that a homomorphism M1 → M2 of left O-modules is an isogeny
if it is injective. We say that an isogeny M1 → M2 is prime to ℘ if its cokernel hasno ℘-torsion.
For a supersingular Drinfeld module E of rank two over Spec κ(℘), we put
FE0 (E) = HomA-mod/κ(℘) (E, E0 ). Here HomA-mod/κ(℘) denotes the set of homomorphisms of schemes of A-modules over Spec κ(℘). The composition law
EndA-mod/κ(℘) (E0 ) × HomA-mod/κ(℘) (E, E0 ) → HomA-mod/κ(℘) (E, E0 )provides FE0 (E) with a structure of a left O-module. Let E 0 be another supersingular Drinfeld module of rank two over Spec κ(℘) and let f : E → E 0 be a morphism ofschemes of A-modules over Spec κ(℘). Then the map FE0 (f ) : FE0 (E 0 ) → FE0 (E)given by the composition with f is a homomorphism of left O-modules.
For a scheme G of A-modules over Spec κ(℘), we denote by G(κ(℘)) the Amodule of scheme theoretic sections of G → Spec κ(℘). We let M0 denote thetorsion A-module E0 (κ(℘)) equipped with a canonical left O-module structure.
Proposition 5.1. Let us consider the following two categories.
(1) The category whose objects are the supersingular Drinfeld modules of rank
two over Spec κ(℘), and whose morphisms are the morphisms of schemesof A-modules over Spec κ(℘).
(2) The category whose objects are the invertible left O-modules, and whosemorphisms are the homomorphisms of left O-modules.
Then the contravariant functor F = FE0 gives an anti-equivalence from the category(1) to the category (2).
Moreover the notions of isogeny and prime-to-℘ isogeny in both categories coincide. If E1 → E2 is a prime-to-℘ isogeny in the category (1), then there is a
canonical isomorphism
Coker[F(E2 ) → F(E1 )] ∼= HomA (Ker[E1 (κ(℘)) → E2 (κ(℘))], M0 ).
of left O-modules. Here Coker (resp. Ker) is the cokernel in the category of left
O-modules (resp. the kernel in the category of A-modules) and the structure of aleft O-module on HomA (Ker[E1 (κ(℘)) → E2 (κ(℘))], M0 ) is the one induced by theleft action of O on M0 .

22

SATOSHI KONDO AND SEIDAI YASUDA

Remark 5.2. The essential part of this proposition is a consequence of the resultsin §2 of [Dr2, p. 161-163]. Let us remark that the description using left ideals is ananalogue of Deuring’s result for elliptic curves and is adapted from [Gek3].
Proof. First we show that for any supersingular Drinfeld module E of rank 2 overSpec κ(℘), the left O-module F(E) = HomA-mod/κ(℘) (E, E0 ) is an invertible left
O-module. Note that any two Drinfeld modules of rank 2 over Spec κ(℘) are isomorphic to each other. Note that any two supersingular Drinfeld modules overSpec κ(℘) are isogenous. Hence F(E) is not zero. Since any isogeny E0 → E induces an injective homomorphism F(E) ,→ F(E0 ) = O of left O-modules, the left
O-module F(E) is isomorphic to a non-zero left ideal of O. Since O is a maximal order of the quaternion division algebra B, any non-zero left ideal of O is aninvertible left O-module. Therefore F(E) is an invertible left O-module.
Let C denote the full subcategory of the category (2) whose objects are thenon-zero left ideals of O. We note that the canonical inclusion functor from C tothe category (2) gives an equivalence of categories. We construct a contravariant
functor G from C to the category (1) which gives a quasi-inverse to the functor F.
In [Gek3, 3.7. Lemma, p. 193], Gekeler constructed, for any non-zero left Oideal A ⊂ O, a supersingular Drinfeld module G(A) over Spec κ(℘). Let us recall
the construction of G(A). Let τ denote the endomorphism of the group scheme
Ga over Spec κ(℘) which induces, for any scheme S over Spec κ(℘), the q-power
endomorphism of the ring Ga (S) = Γ(S, OS ), where q is as in Section 4.1. Forx ∈ κ(℘), we also denote by x the endomorphism of Ga over Spec κ(℘) which
induces, for any scheme S over Spec κ(℘), the endomorphism of the additive group
Ga (S) = Γ(S, OS ) supplied by the multiplication by x. We ﬁx an isomorphism
E0 ∼= Ga as group schemes over Spec κ(℘). By abuse of notation, we denote byτ , x the endomorphisms of E0 as a group scheme over Spec κ(℘) corresponding
to the endomorphisms τ , x, respectively. Let R be the subring of the ring ofendomorphisms of E0 as a group scheme over Spec κ(℘), which is generated by τand by all x ∈ κ(℘). Then R is a non-commutative polynomial algebra over κ(℘)
in one variable τ . The ring R contains no zero divisor and any left ideal of R is
generated by one element. Note that the ring O = EndA-mod/κ(℘) (E0 ) is a subringof R. For a left ideal A ⊂ O, let uA ∈ R denote the unique monic polynomial in τ
satisfying RA = RuA . The structure of a scheme of A-modules over Spec κ(℘) on
E0 gives a ring homomorphism ι0 : A → R, and O equals the commutant of ι0 (A)in R. Let a ∈ A. Since RuA ι0 (a) = RAι0 (a) ⊂ RA = RuA , there exists a uniqueelement ιA (a) ∈ R such that uA ι0 (a) = ιA (a)uA . The map ιA : A → R is a ringhomomorphism. The pair (E0 , ιA ) deﬁnes a Drinfeld module G(A) of rank 2 overSpec κ(℘).
Let A, A0 ⊂ O be two left ideals, and let f : A → A0 be a homomorphism of left
O-modules. We construct a morphism G(f ) : G(A0 ) → G(A) of Drinfeld modules
over Spec κ(℘). There exists a unique element b ∈ B such that the diagram
⊂

⊂

⊂

⊂

A −−−−→ O −−−−→ B




fy
yb
A0 −−−−→ O −−−−→ B,

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

23

where the horizontal maps are the canonical inclusions and the right vertical map is
supplied by the right multiplication by b, is commutative. Since the endomorphism
R ⊗ι0 ,A k → R ⊗ι0 ,A k supplied by the right multiplication by b sends an elementin RA to an element in RA0 , it induces a homomorphism fR : RA → RA0 of
left R-modules. By construction, the homomorphism fR commutes with the rightmultiplication by any element in ι0 (A). Recall that RA = RuA and RA0 = RuA0 .
Let rf ∈ R be the unique element satisfying fR (uA ) = rf uA0 . Then for any a ∈ A
we haverf ιA0 (a)uA0 = rf uA0 ι0 (a) = fR (uA )ι0 (a) = fR (uA ι0 (a))= fR (ιA (a)uA ) = ιA (a)fR (u) = ιA (a)rf uA0 .
Hence we have rf ιA0 (a) = ιA (a)rf for any a ∈ A. Therefore, the element rf ∈ R
gives a homomorphism G(f ) : G(A0 ) → G(A) of Drinfeld modules over Spec κ(℘).
If A00 ⊂ O is another left ideal and if f 0 : A0 → A00 is a homomorphism of left
O-modules, then it is easily checked that G(f 0 ◦ f ) = G(f ) ◦ G(f 0 ). Hence G is a contravariant functor from the category C to the category (1). By [Gek3, 4.3. Theorem,
p. 196], the functor G is essentially surjective.
We show that the composition F ◦ G is naturally isomorphic to the canonicalinclusion functor from C to the category (2). Let A ⊂ O be a left ideal. The imageof the injective homomorphism F ◦ G(A) ,→ F ◦ G(O) = F(E0 ) = O induced by the
inclusion A ⊂ O is equal to the set RA ∩ O which we denote by A0 . The set A0 is aleft ideal of O with A ⊂ A0 and, since RA = RA0 , the homomorphism G(A0 ) → G(A)induced by the canonical inclusion A ⊂ A0 is an isomorphism. Let nrd : B → kdenote the reduced norm. By [Gek3, 3.10. Lemma, p. 194], the ideal of A generatedby {nrd(b0 ) | b0 ∈ A} is equal to the ideal of A generated by {nrd(b0 ) | b0 ∈ A0 }.
Using this, one can check that A ⊗A A℘0 = A0 ⊗A A℘0 as left ideals of O ⊗A A℘0 for
every nonzero prime ideal ℘0 ⊂ A. Hence we have A = A0 . We easily see that theisomorphism
F ◦ G(A) ∼= Image[F ◦ G(A) ,→ F ◦ G(O) = F(E0 ) = O] = A
gives a functorial isomorphism from F ◦ G to the canonical inclusion functor fromC to the category (2).
Therefore the functor F is essentially surjective and for any two supersingularDrinfeld modules E, E 0 of rank two over Spec κ(℘), the map(5.1)

HomA-mod/κ(℘) (E, E 0 ) → HomO (F(E 0 ), F(E))

is surjective. Since both the source and the target of (5.1) are locally free Amodules of rank 4, the map (5.1) is an isomorphism. Hence F gives an equivalenceof categories.
Let ϕ : E1 → E2 be a prime-to-℘ isogeny of supersingular Drinfeld modules ofrank 2 over Spec κ(℘). Let X denote the kernel of ϕ. The exact sequenceϕ

0 → X → E1 −→ E2 → 0of schemes of A-modules over Spec κ(℘) gives an exact sequence
0 → F(E2 ) → F(E1 ) → HomA-mod/κ(℘) (X, E0 )of left O-modules. SinceLϕ is a prime-to-℘ isogeny, we have an isomorphism0
HomA-mod/κ(℘) (X, E0 ) ∼=℘0 HomA-mod/κ(℘) (X, E0 ) ⊗A A℘0 , where ℘ runs over0
the primes of k diﬀerent from ∞, ℘. Let ℘ be a prime of k diﬀerent from ∞, ℘. For

24

SATOSHI KONDO AND SEIDAI YASUDA∞

a scheme G of A-modules over Spec κ(℘), let G[℘0 ] denote the A-module of ℘0 primary torsion elements of G(κ(℘)). Take a ﬁnite subﬁeld L ⊂ κ(℘) such that E0 ,E1 , E2 , and ϕ are deﬁned over L. Then for i = 0, 1, 2, the group Gal(κ(℘)/L) acts∞on the A-module Ei [℘0 ]. By [Yu, Theorem 2, p. 164], we have an isomorphism
∞
∞
(5.2)
F(Ei ) ⊗A A℘0 ∼HomA[Gal(κ(℘)/L0 )] (Ei [℘0 ], E0 [℘0 ])
= lim
−→
0
L

of left O-modules for i = 1, 2, where L0 runs over the ﬁnite subﬁelds of κ(℘)
containing L. By [Yu, EXAMPLE 0, p. 166–167], the right hand side of (5.2) is∞∞∞isomorphic to HomA (Ei [℘0 ], E0 [℘0 ]) = HomA (Ei [℘0 ], M0 ). The A-module M0
is an injective A-module, since it is divisible as an A-module. Hence Coker[F(E2 ) →F(E1 )] ⊗A A℘0 is canonically isomorphic to∞∞∞

HomA (X[℘0 ], E0 [℘0 ]) = HomA (X[℘0 ], M0 ).
Hence Coker[F(E2 ) → F(E1 )] is canonically isomorphic to HomA (X(κ(℘)), M0 ).
¤5.2.2. For a ring L, let Mat
L2 (L) denote the L-algebra of two-by-two matrices withentries in L. Let (k/A)0 = ℘0 6=℘,∞ k℘ /A℘ be the prime-to-℘ part of the torsion A⊕2
module k/A. Then M0 is, as an A-module, isomorphic to the direct sum (k/A)00
0 ⊕2 ∼of two copies of (k/A) . We ﬁx one such isomorphism ξ : (k/A)= M0 . Let0bbb0b0 = Q 00Abetheprime-to-℘
part
of
A
and
put
O
=
O
⊗
A
A A . We℘ 6=℘,∞ ℘⊕2⊕2regard (k/A)0 as the A-module of column vectors. Then (k/A)0 has a canonical00b )-module and the action of Mat2 (Ab ) on (k/A)0 ⊕2 gives
structure of a left Mat2 (A⊕2b0 ) ∼
an isomorphism Mat2 (A= EndA ((k/A)0 ) of rings. The composite
O → EndA (M0 ) ∼= EndA ((k/A)0

⊕2

b0 ),
)∼
= Mat2 (Awhere the isomorphism in the middle is the one induced by the ﬁxed isomorphism
b0 ∼
b0 ).
ξ, induces an isomorphism η : O= Mat2 (ALet E be a supersingular Drinfeld module of rank two over Spec κ(℘). Let I ⊂ A
be a non-zero ideal which is prime to ℘. To each level I structure ϕ : (I −1 /A)⊕2 ∼=E[I] on E, we associate the map
ΦE0 ,ξ (ϕ) : FE0 (E) → O/IO
which is deﬁned as the composite of the following four maps
• the canonical map FE0 (E) = HomA-mod/κ(℘) (E, E0 ) → HomA (E(κ(℘)), M0 ).
• the restriction map HomA (E(κ(℘)), M0 ) → HomA (E[I](κ(℘)), M0 ).
• the isomorphism
HomA (E[I](κ(℘)), M0 ) ∼= HomA ((I −1 /A)⊕2 , M0 )induced by ϕ.
• the inverse of the isomorphism(5.3)
O/IO ∼= HomA ((I −1 /A)⊕2 , M0 )of left O-modules which sends 1 ∈ O/IO to the restriction of the ﬁxed
⊕2
isomorphism ξ to the submodule (I −1 /A)⊕2 ⊂ (k/A)0 .
Then, as a consequence of the anti-equivalence in Proposition 5.1, we have thefollowing.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

25

Corollary 5.3. Let E be a supersingular Drinfeld module of rank two over Spec κ(℘).
Let I ⊂ A be a non-zero ideal which is prime to ℘.
(1) For each level I structure ϕ : (I −1 /A)⊕2 ∼= E[I] on E, the map ΦE0 ,ξ (ϕ) isa surjective homomorphism of left O-modules.
(2) The map ϕ 7→ ΦE0 ,ξ (ϕ) gives a bijection between the set of Drinfeld level I
structures on E and the set of surjective homomorphisms FE0 (E) → O/IO
of left O-modules.
¤b = O ⊗A A.
b Let K0 denote the kernel of the homomorphism
5.2.3. We set O
I××b → (O/IO) . For x ∈ (B⊗k A∞ )× , let Mx denote the intersection B∩(O·xb −1 )(O)∞in B ⊗k A . Then Mx is an invertible left O-module and the composite
b · x−1 ∼b ³ O/IO,
Mx ,→ O
=O
where the injection on the left is the canonical inclusion and the isomorphism inthe middle is the homomorphism supplied by the right multiplication by x, gives asurjection Mx → O/IO.
Fix a supersingular Drinfeld module E0 over Spec κ(℘). Let M0 be as in Propo⊕2 ∼sition 5.1 and ﬁx an isomorphism ξ : (k/A)0= M0 of A-modules. By Proposition 5.1 and Corollary 5.3, the invertible left O-module Mx with the surjection
Mx → O/IO deﬁned above corresponds to a supersingular Drinfeld module Ex ofrank two over Spec κ(℘) with a Drinfeld level I structure ϕx : (I −1 /A)⊕2 ∼= Ex [I]on Ex .
By Proposition 5.1 and Corollary 5.3, the map which sends x ∈ (B ⊗k A)× to thesupersingular Drinfeld module Ex of rank two over Spec κ(℘) with a Drinfeld level
I structure ϕx gives a bijection (depending on E0 and ξ) between the set ΣI,℘ of theisomorphism classes of supersingular Drinfeld modules of rank two over Spec κ(℘)
with a Drinfeld level I structure and the double coset
B × \(B ⊗k A∞ )× /K0I .
Let Σlim,℘ denote the proﬁnite set limI ΣI,℘ where I runs over the non-zero ideals←−
I ⊂ A such that Spec (A/I) contains two or more closed points and that I is prime
to ℘. Passing to the projective limit we have an isomorphism(5.4)

Σlim,℘ ∼= B × \(B ⊗k A∞ )× /(O ⊗A A℘ )×of sets.
Drinfeld [Dr1, §5, D)] introduced the right action of the group GLd (A∞ ) on the
projective limit limI MId , where I runs over the non-zero ideals I ⊂ A (we note←−that this projective limit exists in the category of schemes over Spec A since, whenSpec (A/I) contains two or more closed points, every MId is an aﬃne scheme). We
restrict our attention to the case where d = 2; we have a continuous right action of
the group GL2 (A∞ ) on the set Σlim,℘ .
We let A∞,℘ denote the prime-to-℘-part of A∞ . In this paper, we often restrict
our attention to the subgroup GL2 (A∞,℘ ) ⊂ GL2 (A∞ ). Recall that in Section 5.2.2
b0 ∼b0 ), where A
b0 = Q 00
we have obtained an isomorphism η : O⊗A A= Mat2 (A℘ 6=∞,℘ A℘b from the ﬁxed isomorphism ξ. This induces an isomoris the prime-to-℘-part of A,∼ GL2 (A∞,℘ ).
phism (B ⊗k A∞,℘ )× =

26

SATOSHI KONDO AND SEIDAI YASUDA

Proposition 5.4. The action of GL2 (A∞,℘ ) on the left hand side of (5.4) coincides
with the action of (B⊗k A∞,℘ )× under the isomorphism (B⊗k A∞,℘ )× ∼= GL2 (A∞,℘ )above.
Proof. This follows immediately from the deﬁnition of the action of the group
GL2 (A∞,℘ ) on limI MI and Lemma 5.5 below.
¤←−
Lemma 5.5. Let E be a supersingular Drinfeld module of rank two over Spec κ(℘).
Let I ⊂ A be a non-zero ideal which is prime to ℘. Let M0 be as in Proposition 5.1⊕2
and ﬁx an isomorphism ξ : (k/A)0 ∼= M0 of A-modules.
(1) Let g be an element in Mat2 (A/I). We also denote by g the endomorphismof (I −1 /A)⊕2 supplied by the left multiplication by g. Let X ⊂ (I −1 /A)⊕2
denote the kernel of g. Let I 0 ⊂ A be an ideal containing I such that−1
(I 0 /A)⊕2 ⊂ g(I −1 /A)⊕2 ⊂ (I −1 /A)⊕2 . Let ϕ : (I −1 /A)⊕2 ∼= E[I] bea level I structure on E. Let E 0 be the quotient of E by ϕ(X) and let
f : E → E 0 be the canonical isogeny. Let ηI : Mat2 (A/I) ∼= O/IO be
the isomorphism induced by the isomorphism η. Then the image of the
FE (f )

ΦE ,ξ (ϕ)

composite FE0 (E 0 ) −−−0−−→ FE0 (E) −−−0−−−→ O/IO is equal to the imageof the map O/IO → O/IO supplied by the right multiplication by ηI (g).
−1(2) Let the notation be as in (1). Let ϕ0 : (I 0 /A)⊕2 ∼= E 0 [I 0 ] be the unique00level I -structure on E satisfying the following property: if a ∈ (I −1 /A)⊕2−1
satisﬁes ga ∈ (I 0 /A)⊕2 , then ϕ0 (ga) is equal to the image of ϕ(a) underthe map E[I] → E 0 [I] induced by f . Then the map ΦE0 ,ξ (ϕ0 ) : FE0 (E 0 ) →O/I 0 O is characterized by the following property: if b ∈ FE0 (E 0 ) and if
we take an element c ∈ O/IO such that (ΦE0 ,ξ (ϕ))(FE0 (f )(b)) = cηI (g),then (ΦE0 ,ξ (ϕ))(b) is equal to the image of c under the natural projection
O/IO → O/I 0 O.
Proof. Let G denote the image of E[I] ⊂ E under the isogeny f : E → E 0 in the
category of schemes of A-modules over Spec κ(℘). Then G contains E 0 [I 0 ] and theisogeny f induces an isomorphism fI : E[I]/ϕ(X) ∼= G. We have the commutative
diagram
HomA (E[I](κ(℘)), M0 )
x
(2)


∼
=

−−−−→
(1)

HomA ((I −1 /A)⊕2 , M0 )
x
(3)


∼
=

(5.5)

HomA (E[I]/ϕ(X)(κ(℘)), M0 ) −−−−→ HomA ((I −1 /A)⊕2 /X, M0 )
(4)


(5)

∼
∼
=y
=y(6)
HomA (g((I −1 /A)⊕2 ), M0 )

(8)
y

HomA (G(κ(℘)), M0 )(7)yHomA (E 0 [I 0 ](κ(℘)), M0 )∼=−−−−→ HomA ((I 0(9)−1/A)⊕2 , M0 ),where the maps are deﬁned as follows:• The maps (1) and (4) are the isomorphisms induced by ϕ.
• The map (2) (resp. (3)) is the injective homomorphism induced by the
canonical surjection E[I] → E[I]/ϕ(X) (resp. (I −1 /A)⊕2 → (I −1 /A)⊕2 /X).

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

27• The map (5) (resp. (6)) is the isomorphism induced by the isomorphism fI(resp. the isomorphism (I −1 /A)⊕2 /X ∼= g((I −1 /A)⊕2 ) supplied by the left
multiplication by g).
• The map (7) (resp. (8)) is the homomorphism induced by the inclusion−1
E 0 [I 0 ] ⊂ G (resp. (I 0 /A)⊕2 ⊂ g((I −1 /A)⊕2 )).
• The map (9) is the isomorphism induced by ϕ0 .
It is easily seen from the deﬁnition of the isomorphism ηI that the image of the injection HomA ((I −1 /A)⊕2 /X, M0 ) ,→ HomA ((I −1 /A)⊕2 , M0 ) maps isomorphically
onto the image (O/IO)ηI (g) of the endomorphism O/IO → O/IO supplied by theright multiplication by ηI (g) under the isomorphism O/IO ∼= HomA ((I −1 /A)⊕2 , M0 )in (5.3). Moreover we have the commutative diagram
O/IO
x



∼
=

−−−−→

HomA ((I −1 /A)⊕2 , M0 )
x



∼
=

(O/IO)ηI (g) −−−−→ HomA ((I −1 /A)⊕2 /X, M0 )xηI (g)
y
HomA (g((I −1 /A)⊕2 ), M0 )


y

O/IO


y
O/I 0 O∼=−−−−→ HomA ((I 0−1/A)⊕2 , M0 ),where the maps are deﬁned as follows:• The left vertical map at the top is the inclusion.
• The left vertical map in the middle is the map supplied by the right multiplication by ηI (g).
• The left vertical map at the bottom is the canonical surjection O/IO →O/I 0 O.
• The right vertical maps are the same as those in the diagram (5.5).
• The horizontal map at the top is the isomorphism (5.3).
• The horizontal map in the middle is the isomorphism induced by the isomorphism (5.3).
• The horizontal map at the bottom is the isomorphism (5.3) with I replaced
by I 0 .
Hence the claims follow from the deﬁnition of the map ΦE0 ,ξ (ϕ).

¤

6. Bad reduction of Drinfeld modular curves
The results in this section (to be used in Section 7) are Lemmas 6.4 and 6.5.
The reduction modulo ℘ of the Drinfeld modular curve of level I℘n is described.
This can be regarded as the Drinfeld modular analogue of [Ka-Ma, p.427, Theorem
13.7.6]. The method is classical. We use the Serre-Tate theory and exotic Igusa
curves. The picture of the bad reduction is the same as in [Ka-Ma, p.395].
In Section 6.1, we give a description of the Drinfeld modular curve at a goodprime using the determinant morphism from Section 4.4. Lemma 6.1 says that1
MI/κ(℘)(which is a spectrum of a ﬁeld and is an analogue of the reduction mod p

28

SATOSHI KONDO AND SEIDAI YASUDA

1of the ring of cyclotomic integers Z[ζn ]) is the ﬁeld of constants of MI2 ×MI1 MI/κ(℘),
and the structure morphism is smooth.
The local study of the moduli using Serre-Tate theory is due to Drinfeld [Dr1].
We give the details in Sections 6.2.1 and 6.2.2. In Section 6.2.3, we express the setof connected components and of cusps of the reduction modulo ℘ as a coset space.
The modular curves considered in Section 6.3 are the analogues of the exotic Igusa
curves in the elliptic modular case [Ka-Ma, p.385]. The main results of this section(Lemmas 6.4 and 6.5) are proved in Section 6.3 and 6.4.
We mention that the description of deformation spaces in the case of elliptic
D-sheaves is found in [Bo]. He treats the higher dimensional cases as well.

6.1. Throughout this section we will ﬁx a non-zero prime ideal ℘ ⊂ A. We denoteby A℘ the ℘-adic completion of A. Let κ(℘) denote the residue ﬁeld at ℘ of A. We
ﬁx an algebraic closure κ(℘) of κ(℘).
Let I ⊂ A be a non-zero ideal such that Spec (A/I) contains two or more closeddfor MId ×Spec A Spec L. Let w : MI2 → MI1
points. For an A-algebra L we write MI/L
be the determinant morphism on MI2 (deﬁned in Section 4.4). Let us consider thefollowing cartesian diagram:
1
MI2 ×MI1 MI/κ(℘)−−−−→ MI2


f
w
y
y
1
MI/κ(℘)−−−−→ MI1

where the morphism at the bottom is the canonical closed immersion.
Lemma 6.1. Suppose that I is prime to ℘. Then
(1) The morphism f is smooth.
(2) The ﬁbers of f are geometrically connected.
Proof. Let g be the structure morphism MI1 → Spec A. Then the composite w ◦ g
is the structure morphism MI2 → Spec A hence is smooth outside Spec (A/I) by
Lemma 4.1 (2). The pullback by Spec κ(℘) → Spec A of the morphisms w ◦ g and
g are then smooth, hence so is f which is the pullback of w. The claim (1) follows.
2
The compactiﬁcation M I of MI2 , constructed by Drinfeld [Dr1, §9, p. 585] (seealso [Le] for the details), is a regular scheme which is proper over Spec A. Since the22generic ﬁber M I ×Spec A Spec k is the smooth compactiﬁcation of the curve MI/k,
2

1
the determinant morphism MI2 → MI1 gives a morphism M I ×Spec A Spec k → MI/k.
2This morphism extends uniquely to a morphism w : M I → MI1 by [EGAII, (6.3.9),
2
1(by
p. 119], since M I is regular and MI1 equals the integral closure of A in MI/k
Lemma 4.2). We know from [Dr1, Proposition 9.3(3), p. 587] and [Dr1, §9 A),2
p. 585] that the boundary M I \ MI2 is the disjoint union of a ﬁnite number of2
copies of MI1 . Let us write MI1 = Spec R. The formal completion of M I along2
any of the components of M I \ MI2 is isomorphic to SpfR[[t]] (loc. cit., see also[Le, Chapter 5, Theorem 1.10, p. 72] and the preceding remark). Hence it suﬃces
to prove that each ﬁber of w is geometrically connected. The morphism w is2
proper, since M I → Spec A is proper. From the rigid analytic description of the1
determinant morphism w given in Proposition 4.5, it follows that the MI/k-schemeON THE RATIONAL K2 OF AN ELLIPTIC CURVE

29

2
1
MI/k
is geometrically connected (note that, by Lemma 4.2, the scheme MI/kis the21spectrum of a ﬁeld). This implies that the MI/k-scheme M I/k is also geometrically211connected, and in particular H 0 (M I/k , OM 2 ) = H 0 (MI/k, OMI/k). We can thenI/k¤apply [EGAIII1, Corollaire (4.3.12), p. 134] to obtain the claim (2).

2
6.2. Local description of MI/κ(℘). Let q℘ denote the cardinality of κ(℘). We

ﬁx a prime element π ∈ A℘ . The canonical projection A℘ → κ(℘) has a uniqueleft inverse κ(℘) → A℘ , which identiﬁes A℘ with the ring κ(℘)[[π]] of formal power
series.
6.2.1. Universal deformation space of formal A℘ -modules over κ(℘) with a Drinfeld
level structure: the height one case. We deﬁne a formal A℘ -module Fb1 over A℘ as
b a . The action of a ∈ κ(℘) ⊂ A℘ on Fb1 isfollows. As a formal group, Fb1 = G
given by the power series aX, and the action of π is given by the power series
f1 (X) = πX + X q℘ . We put F1 = Fb1 ⊗A℘ κ(℘). Then F1 is a formal A℘ -module ofheight one over κ(℘). By [Dr1, Proposition 1.6, p. 566], any formal A℘ -module overburb κ(℘) κ(℘). Let A
κ(℘) of height one is isomorphic to F1 ⊗℘ denote the completionof the strict henselization of A℘ . Then it can be checked easily that the formalburb A℘ Ab
A℘ -module Fb1 ⊗℘ is the universal deformation of F1 ⊗κ(℘) κ(℘).
b κ(℘) κ(℘). For r ≥ 0 let g1,r =Let D1,n be the universal deformation ring of F1 ⊗f1 ◦ · · · ◦ f1 be the r-th iteration of f1 . Then for n ≥ 1 the ring D1,n is isomorphicburto A℘ [[x]]/(g1,n (x)/g1,n−1 (x)) (cf. [Dr1, proof of Proposition 4.3, p. 573]). Let us
note that the reduction modulo π of D1,n is isomorphic to κ(℘)[[x]]/(xq℘ −q℘ ).
Let I ⊂ A be a non-zero ideal, which is prime to ℘, such that Spec (A/I) contains
two or more closed points. We set In = I℘n . Let Pn denote the set of A-submodulesof (℘−n /A)⊕2 which is free of rank one over A/℘n .
2
Let z ∈ MI/κ(℘)(κ(℘)) be a κ(℘)-rational point which corresponds to an orn

n−1

dinary Drinfeld module Ez over κ(℘). That is, if we regard z as a morphism
2
Spec κ(℘) → MI/κ(℘), then Ez is the pullback by z of the universal Drinfeld module2
over MI/κ(℘). Let MI2 /κ(℘),z = Spec κ(℘) ×M 2
nI/κ(℘)

MI2 /κ(℘) be the ﬁber productn2of z and the canonical morphism MI2 /κ(℘) → MI/κ(℘). The following descripn

c2tion of the formal completion M
of MI2 /κ(℘) along MI2 /κ(℘),z is due toI /κ(℘),znnn

Drinfeld [Dr1, §5 C) Proposition 5.4, p. 576]. If we ﬁx an isomorphism of the forc2
mal A℘ -module associated to Ez to F1 , then M
is canonically isomorphic
In /κ(℘),z`
Spf (D1,n /(π)[[y]]) of copies of Spf (D1,n /(π)[[y]]) ∼to the disjoint union=Q∈Pn

Spf κ(℘)[[x, y]]/(xq℘ −q℘
n

n−1

).

6.2.2. Universal deformation space of formal A℘ -modules over κ(℘) with a Drinfeld
level structure: the height two case. We deﬁne a formal A℘ -module Fb2 over the ringb a . The action ofA℘ [[t]] of formal power series as follows. As a formal group, Fb2 = Gba ∈ κ(℘) ⊂ A℘ on F2 is given by the power series aX, and the action of π is given
2by the power series f2 (X) = πX + tX q℘ + X q℘ . We put F2 = Fb2 ⊗A℘ [[t]] κ(℘). Then
F2 is a formal A℘ -module of height two over κ(℘). By [Dr1, Proposition 1.6, p. 566],b κ(℘) κ(℘).
any formal A℘ -module over κ(℘) of height two is isomorphic to F2 ⊗

30

SATOSHI KONDO AND SEIDAI YASUDA

bur [[t]] is the universal deformationb A℘ [[t]] A
Lemma 6.2. The formal A℘ -module Fb2 ⊗℘b κ(℘) κ(℘).
of F2 ⊗
Proof. By [Dr1, Proposition 4.2, p. 570], the universal deformation ring of theburformal A℘ -module F2 equals A℘ [[t1 ]]. The universality gives the homomorphism
ur
ur
b
b
ϕ : A℘ [[t1 ]] → A℘ [[t]]. Since the coeﬃcient of X in f2 is π, the homomorphism ϕbur -algebra. It suﬃces to prove that the image ϕ(t1 ) is a
is a homomorphism of A℘eburtopological generator of A
℘ [[t]]. Let ΛA℘ (resp. ΛA℘ ) be the graded ring ΛO (resp.
e O ) deﬁned in [Dr1, §1, p. 563] for O = A℘ . In [Dr1, Proposition 1.4, p. 565],
Λ
Drinfeld shows that ΛA℘ equals the polynomial ring with generators g1 , g2 , · · · ∈ΛA℘ such that deg gi = i. In view of his construction of the gj ’s, we may choosee A equals the element u in [Dr1,gq℘ −1 in such a way that the image of gq℘ −1 in Λ℘
bur
Proposition 1.3 (2), p. 564] for n = q℘ . By the construction of A
℘ [[t1 ]], there existsurb℘ [[t1 ]] such that the image of gq −1
a canonical ring homomorphism ΛA℘ → A℘
equals t1 . This implies that ϕ(t1 ) = −t/(1 − π q℘ −1 ) + O(t2 ).
¤b κ(℘) κ(℘)
The following description of the universal deformation ring D2,n of F2 ⊗of level n is due to Drinfeld [Dr1, proof of Proposition 4.3, p. 573]. For n = 1, it isgiven by
2
q℘
−1

2
q℘
−1

θ
− θ1
q −1
q℘ −1
bur
D2,1 ∼+ θ1℘ , t + 2q℘ −1=A℘ [[t, θ1 , θ2 ]]/(π + tθ1
q −1 ).
θ2− θ1℘
2

For r ≥ 0 let g2,r = f2 ◦ · · · ◦ f2 be the r-th iteration of f2 . Then for n ≥ 1 the ring
D2,n is isomorphic to
D2,1 [[y1 , y2 ]]/(g2,n−1 (y1 ) − θ1 , g2,n−1 (y2 ) − θ2 ).
2
Let z ∈ MI/κ(℘)(κ(℘)) be a κ(℘)-rational point which corresponds to a super-

singular Drinfeld module Ez over κ(℘). That is, if we regard z as a morphism
2
Spec κ(℘) → MI/κ(℘), then Ez is the pullback by z of the universal Drinfeld mod2
. Let MI2 /κ(℘),z = Spec κ(℘)×M 2
ule over MI/κ(℘)
nI/κ(℘)

MI2 /κ(℘) be the ﬁber productn2of z and the canonical morphism MI2 /κ(℘) → MI/κ(℘). The following description ofnc2
the formal completion M
of MI2 /κ(℘) along the ﬁber MI2 /κ(℘),z of z is due
I /κ(℘),znnn

to Drinfeld [Dr1, §5 C) Proposition 5.4, p. 576]. If we ﬁx an isomorphism of thec2formal A℘ -module associated to Ez to F2 , then M
is canonically isomorphic
I /κ(℘),zn

to Spf (D2,n /(π)).
6.2.3. Let I ⊂ A be a non-zero ideal, which is prime to ℘, such that Spec (A/I)
2
contains two or more closed points. A closed point in MI/κ(℘)
is called ordinary(resp. supersingular) if the corresponding Drinfeld module is ordinary (resp. su2. Let
persingular). We denote by ΣI,℘ the set of supersingular points of MI/κ(℘)
2
SI,℘ denote the set of connected components of MI/κ(℘). There is a canonical map
ΣI,℘ → SI,℘ sending a point to the connected component to which it belongs.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

31

We ﬁx an algebraic closure k ∞ of k∞ . For an integer n ≥ 0, let CuspI,℘n denote2
the set of cusps of MI℘, that is, the set of the closed points of the complement
n /k
∞

2
of MI℘
in its smooth compactiﬁcation.
n /k∞Let Slim,℘ (resp. Cusplim,℘n ) denote the proﬁnite set limI SI,℘ , (resp. limI CuspI,℘n )←−←−where I runs over the non-zero ideals I ⊂ A such that such that Spec (A/I) contains
two or more closed points and that I is prime to ℘. There is a canonical morphismΣlim,℘ → Slim,℘ of proﬁnite sets.
The right action the group GLd (A∞ ) on the projective limit limI MId , where I←−
runs over the non-zero ideals I ⊂ A gives a continuous right action of the group
GL2 (A∞ ) on the set Slim,℘ , and the canonical morphism Σlim,℘ → Slim,℘ describedin the previous paragraph is GL2 (A∞,℘ )-equivariant.

Lemma 6.3. Let T ⊂ GL2 (resp. N ⊂ GL2 ) denote the diagonal torus (resp. thesubgroup of upper triangular unipotent matrices) of GL2 .
(1) There are canonical isomorphisms
SI,℘
CuspI,℘n∼b× → (A/I)× ),= k × \(A∞ )× /Ker(A∞∼= T (k)N (A )\GL2 (A∞ )/KI℘n .

(2) There are canonical isomorphisms
Slim,℘
Cusplim,℘n∼= k × \(A∞ )× /A×℘,∼= T (k)N (A∞ )\GL2 (A∞ )/(1 + ℘n Mat2 (A℘ ))of GL2 (A∞,℘ )-sets. Here GL2 (A∞,℘ ) acts on (A∞ )× /A×℘ via the determinant det : GL2 (A∞,℘ ) → (A∞,℘ )× .
(Recall the deﬁnition of K∗ from Section 4.3.2.)
Proof. The set of cusps CuspI,℘n is isomorphic to the double coset
GL2 (k)\(GL2 (A∞ )/KI℘n × P1 (k)) ∼= T (k)N (k)\GL2 (A∞ )/KI℘n(see, for example, [Gek2, (1.2.5), p. 354]). The claim for the cusps follows since
N (k) is dense in N (A∞ ).
We prove the claim for SI,℘ . By Lemma 6.1, the set SI,℘ is canonically iso1
morphic to the set of connected components of MI/κ(℘). Since MI1 ×Spec A UI is
ﬁnite étale over UI , it is also isomorphic to the set of connected components of
MI1 ×Spec A Spec k ∞ By the adelic description (4.2) of (MI1 ×Spec A Spec k℘ )an , it isalso isomorphic to the double coset
b× → (A/I)× ).
k × \(A∞ )× /Ker(AThis proves (1). Taking the projective limit, we have (2).

¤6.3. Fix an integer n ≥ 1. For a non-zero ideal I ⊂ A such that Spec (A/I)
contains two or more closed points and that I is prime to ℘, we put In = I℘n . Let
us describe the reduction at ℘ of the moduli scheme MI2n .
For Q ∈ Pn (deﬁned in Section 6.2.1), we let MI2n /κ(℘),Q ⊂ MI2n /κ(℘) denote theclosed subscheme which classiﬁes the Drinfeld modules of rank two with a Drinfeld
level In structure φ such that Q ⊂ Ker φ. This is an analogue of the Igusa curve.
We set MI2 /κ(℘),Q = MI2n /κ(℘),Q ×Spec κ(℘) Spec κ(℘).
n

32

SATOSHI KONDO AND SEIDAI YASUDA

Lemma 6.4.
(1) The underlying topological space of MI2n /κ(℘) equals the unionof the underlying topological spaces of the MI2n /κ(℘),Q ’s.
(2) For Q, Q0 ∈ Pn with Q 6= Q0 , the scheme MI2n /κ(℘),Q intersects MI2n /κ(℘),Q0
only at supersingular points.
(3) The scheme MI2n /κ(℘),Q is a smooth curve over Spec κ(℘).
2(4) The morphism MI2n /κ(℘),Q → MI/κ(℘)
is ﬁnite, ﬂat, surjective, of con2
stant degree and is totally ramiﬁed at every supersingular point on MI/κ(℘).
This also induces a bijection between the set of connected components of
2.
MI2 /κ(℘),Q and that of MI/κ(℘)
n

Proof. We will prove the following four statements for every closed point x ∈2
MI/κ(℘):(1)0 The ﬁber MI2 /κ(℘),x of MI2 /κ(℘) at x is non-empty and is the union of thenn

MI2 /κ(℘),Q,x ’s.
n(2)0 If x is an ordinary point, then MI2 /κ(℘),x is the disjoint union of thenMI2 /κ(℘),Q,x ’s.
n(3)0 The scheme MI2 /κ(℘),Q is smooth over Spec κ(℘) in a neighborhood ofn

MI2 /κ(℘),Q,x .
n2(4)0 If x is a supersingular point, then the morphism MI2 /κ(℘),Q → MI/κ(℘),Qn
is totally ramiﬁed at x, whose ramiﬁcation index does not depend on the
choice of x.
It is clear that the assertions (1), (2) and (3) follow from (1)0 , (2)0 , and (3)0 . Let2
us denote by f the morphism MI2n /κ(℘),Q → MI/κ(℘)
in the statement of (4). Then
2
2
f is ﬁnite ﬂat since MI℘n → MI is ﬁnite ﬂat. By (1), we see that f is surjective.
The remaining assertions in (4) follow from (4)0 and the existence of a supersingular2
point on each connected component of MI/κ(℘).
Let E → x denote the Drinfeld module of rank two over x corresponding to thegeometric point x. Then the ℘n -torsion subgroup of the A-module E(x) is a freeA/℘n -module of rank ≤ 1. Hence for any A-module homomorphism (In−1 /A)⊕2 →E(x), its kernel contains a submodule isomorphic to A/℘n . We regard this as anelement Q ∈ Pn . This implies (1)0 .
The group GL2 (A/℘n ) acts both on Pn and on MI2n /κ(℘) . The action of g ∈GL2 (A/℘n ) on MI2 /κ(℘) maps MI2 /κ(℘),Q isomorphically onto MI2 /κ(℘),gQ . Sincennn
the group GL2 (A/℘n ) acts transitively on Pn ∼’s are= P1 (A/℘n ), all the M 2
In /κ(℘),Q2
isomorphic as schemes over MI/κ(℘). Hence it suﬃces to show that (3)0 and (4)0hold for Q = Q0 , where Q0 is the second direct summand ℘−n /A of (℘−n /A)⊕2 .
It follows from the description in Section 6.2.1 that the completion of the local
ring of MI2 /κ(℘),Q at an ordinary point equals
0n(D1,n ⊗Abur κ(℘))[[y]]/(x) ∼= κ(℘)[[y]].
℘0This proves (2) .
The claim (3)0 for an ordinary point z immediately follows from the argumentin the proof of (2)0 . Suppose that z is a supersingular point. It follows from theON THE RATIONAL K2 OF AN ELLIPTIC CURVE

33

description in Section 6.2.2 that the coordinate ring of the completion of MI2 /κ(℘),Qn0along the ﬁber of z equals
q 2 −q℘(D2,n ⊗Abur κ(℘))/(y2 ) = κ(℘)[[t, θ1 , y1 ]]/(t + θ1℘℘, g2,n (y1 ) − θ1 ).

From this description, we easily see that the canonical homomorphism κ(℘)[[y1 ]] →(D2,n ⊗Abur κ(℘))/(y2 ) is an isomorphism. Hence MI2 /κ(℘),Q is smooth over Spec κ(℘)
℘

n

0

2
in a neighborhood of MI2 /κ(℘),Q ,z , and MI2 /κ(℘),Q → MI/κ(℘)
is totally ramiﬁed
n

0

n

0

at z. This proves (3)0 .
Let z be a supersingular point. Then, using the notations in the proof of (3)0 ,2
the ring homomorphism from the completion of the local ring of MI/κ(℘)
at z tothe coordinate ring of the completion of MI2 /κ(℘),Q along the ﬁber of z is then0homomorphismκ(℘)[[t]] → (D2,n ⊗ bur κ(℘))/(y2 ) ∼= κ(℘)[[y1 ]].
A℘

This homomorphism is totally ramiﬁed and its ramiﬁcation index is independentof the choice of z. This proves (4)0 .
¤6.4. Fix a non-zero ideal I ⊂ A, which is prime to ℘, such that Spec (A/I) contains
two or more closed points. Let n ≥ 0 be an integer. We write In = I℘n . Let us2
consider the compactiﬁcation M In of MI2n constructed by Drinfeld [Dr1, §9, p. 585].
2

2

Lemma 6.5. For Q ∈ Pn , let M In /κ(℘),Q ⊂ M In denote the closure of MI2n /κ(℘),Q .
Then
2(1) The scheme M In /κ(℘),Q is a projective smooth curve over κ(℘).
22(2) For Q0 ∈ Pn with Q 6= Q0 , the schemes M In /κ(℘),Q and M In /κ(℘),Q0 do not2intersect at a point in the boundary M In \ MI2n .
2

Proof. We note that, to prove (1), it suﬃces to show that M In /κ(℘),Q is non-singular2

at the boundary M In /κ(℘),Q \ MI2n /κ(℘),Q .
2

2

Since MI2n /κ(℘),Q is a smooth curve and M In is regular, the closure M In /κ(℘),Q2

is reduced. Hence both (1) and (2) follow if we prove that (M In /κ(℘) )red is regularat the boundary.
Let us write MI1 = Spec R. As noted in the proof of Lemma 6.1, the boundary
2
M In \ MI2n is the disjoint union of a ﬁnite number of copies of Spec R, and for each2

2component of M In \ MI2n , the coordinate ring of the completion of M In along it isisomorphic to R[[t]]. The claim follows since (R ⊗A κ(℘)[[t]])red is regular.
¤
7. Integrality
The main result of this section is Corollary 7.4. We use this in the form Proposition 7.6 in the proof of Proposition 9.1. This is the only result to be used in othersections.
We give in this section the Drinfeld modular analogue of the result of Beilinson
on integrality in the form presented by Schappacher and Scholl [Sc-Sc, Section 7,
p. 298-302]. Let us brieﬂy recall (a part of) their result. Let N ≥ 1 be an integer,
which we assume to be divisible by at least two distinct primes. In his work on

34

SATOSHI KONDO AND SEIDAI YASUDA

the K2 of a modular curve, concerning his conjectures on L-functions, Beilinson
constructs what are called Beilinson elements in the rational K2 of the compactiﬁed modular curves X(N ) over Q where N means the level. The method is,
ﬁrst consider the image of the symbol map O(Y (N ))×⊗2 → K2 (Y (N )) for (open)
modular curve Y (N ), then to use Bloch’s method (See the proof of Lemma 9.3;
the crucial ingredient is the Drinfeld-Manin theorem on the ﬁniteness of the cuspidal divisor class group.) to obtain elements in K2 (X(N ))Q . The integralityof elements in K2 (X(N ))Q means that they lie in the kernel of the boundary map⊕p ∂ pK2 (X(N ))Q −−−→ ⊕p G1 (X(N )p )Q where the subscript p means the reduction modulo p of the regular model over Z.
As the G1 (or equivalently K1 ) of a projective smooth curve over a ﬁnite ﬁeld is
torsion (see Lemma 7.5), we only need to consider those primes p dividing the level
N . At those primes p dividing the level N , one requires the study of the reductionof modular curves (we have presented the Drinfeld modular analogue in Section 6),
and some facts from the theory of automorphic forms.
In Section 7.1, we give a basic result (Lemma 7.1), which is an application of theresults of previous sections. In Section 7.2, we give the analogue of [Sc-Sc]***. In
Section 7.3, we put the result of this section in the form to be used in later section.
This requires some easy computation of the rational G-group of a curve over a ﬁnite
ﬁeld and some diagram chasing.
Let I ( A be a non-zero ideal such that Spec (A/I) contains two or more closed
points. For a scheme X, we let O(X) = H 0 (X, OX ).
7.1. Let M be an abelian group. Let us recall that we have introduced thetwo sets ΣI,℘ and SI,℘ in Section 6.2.3. We write M [ΣI,℘ ], M [SI,℘ ], for thegroups of M -valued functions on the sets ΣI,℘ , SI,℘ , respectively. The canonical map ΣI,℘ → SI,℘ induces a homomorphism γI : M [SI,℘ ] → M [ΣI,℘ ]. We set
C(Σlim,℘ , M ) = limI M [ΣI,℘ ] where I runs over all non-zero ideals of A prime to−→℘ such that Spec (A/I) contains two or more closed points, and deﬁne the abeliangroups C(Slim,℘ , M ) and C(Cusplim,℘ , M ), and the homomorphism γlim in a similar manner. The action of GLd (A∞ ) on limI MI2 which we have recalled in Sec←−tion 6.2.3 induces actions of the group GL2 (A∞,℘ ) on the sets C(Slim,℘ , M ) andC(Cusplim,℘ , M ), and the map γlim is GL2 (A∞,℘ )-equivariant.
Lemma 7.1. The cokernel of γlim : C(Slim,℘ , C) → C(Σlim,℘ , C) is a direct sumof irreducible admissible representations of GLd (A∞,℘ ). Each direct summand is
cuspidal automorphic, that is, isomorphic to the A∞,℘ -component of a cuspidal
irreducible automorphic representations of GLd (A).
Proof. By the isomorphisms in Proposition 5.4 and in Lemma 6.3, we can regard
the canonical map Σlim,℘ → Slim,℘ as a GL2 (A℘,∞ )-equivariant map
B × \(B ⊗k A∞ )× /(O ⊗A A℘ )× → k × \(A∞ )× /A×℘
where the group GL2 (A℘,∞ ) acts on the left hand side via the isomorphism (B ⊗kA∞,℘ )× ∼= GL2 (A∞,℘ ) in Section 5.2.2.
Observe that any GL2 (A℘,∞ )-equivariant map from B × \(B ⊗k A∞ )× /(O ⊗AA℘ )× to k × \(A∞ )× /A×℘ equals the composite
B × \(B ⊗k A∞ )× /(O ⊗A A℘ )× −−→ k × \(A∞ )× /A×→ k × \(A∞ )× /A×
℘ −
℘
nrd

α

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

35

for some (A∞,℘ )× -equivariant isomorphism α of sets, where the map nrd is themap induced by the reduced norm B → k on B. This shows that the image of γlim
is equal to the C-vector space of locally constant C valued functions f on the set
B × \(B ⊗k A∞ )× /(O ⊗A A℘ )× such that f is the pullback of a function on the set
k × \(A∞ )× /A×℘ under the map nrd.
Observe that the reduced norm induces an isomorphism(B ⊗k A)×ab −−→ A× .
nrd

Hence an automorphic representation of (B ⊗k A)× is not one dimensional if andonly if it occurs in the cokernel of the pullback homomorphism by the reduced norm
S(k × \A× ) → S(B × \(B ⊗k A)× ).
Therefore the claim follows from the following result which is stated in [Ja-La,
Section 14, p. 470] and is worked out by Gelbart and Jacquet [Ge-Ja, Theorem
8.3, p. 245]: for any automorphic representation π = ⊗0℘0 π℘0 (here ℘0 runs over the(possibly the inﬁnity) places of k) of (B ⊗k A)× which is not one dimensional, its“non-(℘, ∞)” component π ℘,∞ = ⊗0℘0 6=℘,∞ π℘0 is isomorphic to the “non-(℘, ∞)”component of a cuspidal automorphic representation of GL2 (A). Here we regard
π ℘,∞ as a representation of GL2 (A℘,∞ ) via the isomorphism (B ⊗k A℘,∞ )× ∼=GL2 (A℘,∞ ) in Section 5.2.2.
¤
7.2. Recall from Lemma 6.4 that the set of irreducible components of MI2n /κ(℘) isisomorphic to Pn where n ≥ 1. Let Q ∈ Pn .
Let MI2,ord
denote the ordinary locus of MI2n /κ(℘),Q , that is, the complementn /κ(℘),Qof supersingular points of MI2n /κ(℘),Q .
In what follows, we consider the two dimensional scheme[
MI2,Q= MI2n /A℘ \MI2n /κ(℘),Q0 .
n
Q0 ∈Pn , Q0 6=Q

We have MI2,Q
×Spec A℘ Spec κ(℘) = MI2,ordand MI2,Q
×Spec A℘ Spec k℘ =nnn /κ(℘),Q
2
MIn /k℘ . The localization sequence induces a homomorphism
2,ord∂In ,Q : K2 (MI2n /k℘ ) → K1 (MI2,ord)∼= O(MIn /κ(℘),Q )× .
n /κ(℘),Q

Here the last isomorphism follows from [Ba-Mi-Se, Corollary 4.3 a), p. 95] in viewof [Mi, p.27-28].
7.2.1. Take a prime element π ∈ A℘ . Let us consider the compositeµeIn ,Q : O(MI2n /k )× ,→ O(MI2n /k℘ )×
{ ,π}

∂I

n
−−−→ K2 (MI2n /k℘ ) −−−−→ O(MI2,ord)× .
n /κ(℘),Q,Q

Lemma 7.2. For u ∈ O(MI2n /k )× , the function µeIn ,Q (u), restricted to a connectedcomponent of MI2,ord, has the same order of pole or zero at each supersingularn /κ(℘),Q
point.
Proof. Using Lemma 6.4(4), we identify the set of supersingular points (resp. connected components) of MI2n /κ(℘),Q with ΣI,℘ (resp. SI,℘ ). An element in the image

36

SATOSHI KONDO AND SEIDAI YASUDA

of γI : Z[SI,℘ ] → Z[ΣI,℘ ] (deﬁned in Section 7.1) is a constant function on ΣI,℘ foreach connected component. We show that the image of the composite map
µ
eI ,Q

n
O(MI2n /k )× −−−−→ O(MI2,ord)× −−→ Z[ΣI,℘ ]n /κ(℘),Q

div

lies in the image of γI : Z[SI,℘ ] → Z[ΣI,℘ ].
Consider the composite with the quotient map, and extend the scalars:O(MI2n /k )× → C[ΣI,℘ ]/C[SI,℘ ].
We prove that this map is zero.
We write the above map as the composite
O(MI2n /k )× → O(MI2,ord)× ⊗Z Q → C[ΣI,℘ ]/C[SI,℘ ]n /κ(℘),Q(7.1)

where the ﬁrst map is µeIn ,Q followed by the extension of scalars. By Lemma 4.1(2),MI2n /k is a smooth curve over k, and it was shown in the proof of Lemma 6.1 that it
is connected. Let k(In ) denote its ﬁeld of constants. Then it can be checked easily
that the ﬁrst map of (7.1) factors through the quotient O(MI2n /k )× /k(In )× .
Let J 0 ⊂ J ⊂ A be two non-zero ideals such that Spec(A/J) contains two or
more closed points. Assume J is prime to ℘.
2
The level lowering morphism (restricted over Spec A℘ ) MJ20 /A℘ → MJ/A
is ﬁ℘
nite ﬂat by Lemma 4.1(3). Considering the moduli problem, we see that MJ2,Q
=
0
n

2
MJ2,Q
×MJ/A
MJ20 /A℘ , so that, as a base change, the level lowering morphism
n
℘

MJ2,Q
→ MJ2,Qis also ﬁnite ﬂat. Since MJ2,ord
00 /κ(℘),Q is canonically isomorphic tonnn

the ﬁber product MJ2,ord
×MJ2 MJ20 , the diagramn /κ(℘),Q{ ,π}2
O(MJ/k)×↓−−−−→

O(MJ20 /k )×

−−−−→

{ ,π}

∂J ,Q

K2 (MJ2n /k℘ )
↓

−
−−n−−
→

K2 (MJ20 /k )

−
−−n−−
→

n

℘

∂ 0
J ,Q

O(MJ2,ord)×n /κ(℘),Q↓→

C[ΣJ,℘ ]/C[SJ,℘ ]↓×
O(MJ2,ord
0 /κ(℘),Q )→C[ΣJ 0 ,℘ ]/C[SJ 0 ,℘ ]n

is commutative. We regard the set of ideals J’s as above as a directed set by the
inclusion of ideals. Passing to the inductive limit (of row in the diagram above)with respect to J (transition maps are the vertical arrows in the diagram, which isa pullback for each term) we obtain a homomorphismµlim,Q :lim O(MJ2n /k )× /k(Jn )× → lim O(MJ2,ord)× ⊗Z Qn /κ(℘),Q−→−→

J, ℘-JJ, ℘-J→ C(Σlim,℘ , C)/C(Slim,℘ , C).
It is GL2 (A∞,℘ )-equivariant by Proposition 5.4. Let KIn ⊂ GL2 (A∞ ) be as in∞,℘
Section 4.3.2. Let K℘). It suﬃces to prove that the image of
In = KIn ∩ GL2 (A℘µlim,Q is zero, since by taking the KIn -invariants, one obtains the claim.
The group O(MJ2n /k )× /k(Jn )× is canonically regarded as a subgroup of theabelian group Z[CuspJ,℘n ]. Passing to the inductive limit, we have an injective
homomorphism
O(MJ2n /k )× /k(Jn )× → C(Cusplim,℘n , Z),from which we see that Image µlim,Q ⊗Z C is isomorphic to a subquotient of thevector space C(Cusplim,℘n , C).
We prove a lemma, which is a consequence of Lemma 6.3.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

37

Lemma 7.3. Let V be an irreducible representation of GL2 (A∞,℘ ), which is asubquotient of C(Cusplim,℘n , C). Let π be an irreducible cupidal automorphic representation of GL2 (A). Then HomGL2 (A∞,℘ ) (V, π) = 0.
Proof of Lemma 7.3. Let V → π be an element of HomGL2 (A∞,℘ ) (V, π) and consider
a cusp form f which belongs to the image of this map. From the deﬁnition of a cusp
form (see, for example, [Co, p.18, Deﬁnition 2.4]), the integral over N (k)\N (A) of
f is zero. By the adelic description of Cusplim,℘n in Lemma 6.3, and using the factthat N (A∞ )N (k) ⊂ N (A) is dense, we see that f is invariant under the action of
N (A∞ ). This implies f = 0. This proves the claim.
¤
By Lemma 7.1, any irreducible subquotient of the target C(Σlim,℘ , C)/C(Slim,℘ , C)of µlim,Q is cuspidal automorphic. Hence µlim,Q ⊗Z C is zero. This completes theproof of Lemma 7.2.
¤
Lemma 6.5 together with Lemma 7.2 implies the following corollary :Corollary 7.4. Let x ∈ K2 (MI2n /k ) be an element which lies in both the imageof the symbol map O(MI2n /k )× ⊗Z O(MI2n /k )× → K2 (MI2n /k ) and the kernel of2the boundary map K2 (MI2n /k ) → K1 (M In /k \ MI2n /k ) ⊗Z Q. Then for any Q ∈ Pn
(deﬁned in Section 6.2.1), the element ∂In ,Q (x) ∈ O(MI2,ord)× is of ﬁnite order.
n /κ(℘),Q
Proof. The argument here follows that given in [Sc-Sc, Section 7.3, p. 300].
PN
Write x = i=1 {u1i , u2i } with ui1 , u2i ∈ O(MI2n /k )× for some positive integer
N . We ﬁx Q ∈ Pn . Then the special ﬁber of MI2,Q
is a divisor and deﬁnes anvaluation on the ring O(MI2n /k℘ ). In the localization of O(MI2n /k℘ ) with respect
to the valuation above, we may write u1i = v1i $a1i and u2i = v2i $a2i so that theorder of v1 and of v2 are zero with respect to the valuation and $ is an element with
a
PN
v 2i
valuation 1. Then since ∂In ,Q (x) = i=1 (−1)a1i a2i v1i
a1i , we only need to consider
2ithe case N = 1, a11 = 0 and a21 = 1.
From Lemma 7.2, we see that the restriction of ∂In ,Q (x) to a connected component has the same number of pole of zero at each supersingular point. From theassumption that x lies in the kernel of the boundary map together with Lemma 6.5,
it follows that ∂In ,Q (x) has no zero or pole at the cusps. Hence ∂In ,Q (x) on eachconnected component is a constant function, so that it is of ﬁnite order since κ(℘)
is ﬁnite.
¤
7.3. We present the result of Sections 7.1 and 7.2 in the form to be used in Proposition 9.1. The proof only requires some diagram chasing. We begin with a lemma.
Lemma 7.5. Let Z be a projective smooth curve over a ﬁnite ﬁeld F. Then G1 (Z)is torsion.
Proof. We have a localization sequenceM(1) M
G0 (κ(z))
G1 (κ(z)) → G1 (Z) → G1 (F(Z)) −−→zz

where z runs over all closed points of Z, and F(Z) is the function ﬁeld of Z. Sinceκ(z) is ﬁnite, G1 (κ(z)) is torsion. It is known that G1 (F(Z)) = F(Z)× and the map(1) is given by the order of the function f ∈ F(Z)× at each z. Since Z is projective,
the kernel of the map (1) must be torsion. This proves the claim.
¤

38

SATOSHI KONDO AND SEIDAI YASUDA

Proposition 7.6. Let I ⊂ A be an ideal such that Spec(A/I) contains two or more
2
closed points. Let x ∈ K2 (MI/k) be an element which lies in both the image of the
2×2
2)× → K2 (MI/k) and the kernel of the boundary
symbol map O(MI/k ) ⊗Z O(MI/k222
map K2 (MI/k) → K1 (M I/k \ MI/k) ⊗Z Q. Let ℘ ⊂ A be a nonzero prime ideal.
22Then ∂℘Q (x) = 0 where ∂℘Q : K2 (MI/k)Q → G1 (MI/κ(℘))Q is the boundary map.
2

Proof. Suppose ℘ is prime to I. Then M I/κ(℘) is a smooth projective curve byLemma 4.1(2) and the proof of Lemma 6.5. Hence ∂℘Q (x) = 0 since the target
space is zero by Lemma 7.5.
Suppose ℘ divides I. Consider the following diagram:O(MI/k )×⊗2 ⊗Z Q
(1)

2

K2 (M I/k )Q(4)/ G1 (M 22
I/k \ MI/k )Q

(3)0

(3)

²

2

K2 (M I/k℘ )Q∂℘Q²
2/ K2 (MI/k
)Q

(2)

²

2
G1 (M I/κ(℘) )Q²
2/ K2 (MI/k
)Q

(5)

℘

∂℘Q

²
2
/ G1 (MI/κ(℘)
)Q

(6)

(7)

L

²
2
G
(M
Q 1
I/κ(℘),Q )Q .

Here the map (1) is the symbol map, the map (2) (5) (6) and (7) are restriction
maps (pullback by an open immersion), the map (4) is the boundary map in thelocalization sequence, and the map (3) and (3)0 are base change maps. We use ∂℘Q ,by abuse, for the boundary maps. Note that the diagram is commutative.
0
0 n
0
Deﬁne
L I and n by I = I ℘ where I is prime to ℘. Then the composite (7)◦∂℘Q
equals Q ∂In0 ,Q where ∂In0 ,Q is as deﬁned above.
The mapL(6) (resp. (7)) is injective since the kernel is the image of the pushforward from p G1 (p)Q where p runs over the cusps (resp. supersingular points).
2

Thus to prove the proposition, it suﬃces to show that the image of x ∈ K2 (M I/k )Q
by the composite (7)∂℘Q (3)0 (2) is zero when (2)(x) lies in the image of (1) and thekernel of (4). This is exactly the claim of Corollary 7.4. This ﬁnishes the proof. ¤8. Regulator
In this section, we reproduce the results of our other paper ([Ko-Ya]) in the form
to be used in this article. Let us brieﬂy comment on the contents of [Ko-Ya]. In thedpaper [Ko-Ya] we consider the rational d-th K-group of the moduli MI/kover k ofDrinfeld modular variety of rank d level I. We construct a map called regulator to
the space of (Q-valued) automorphic forms (with certain conditions at inﬁnity) andprove the surjectivity. The case of rank two, in a more precise form, is quoted below
as Lemma 8.1. The contents of Sections 7 and 9 (especially Proposition 9.1) may

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

39

2

be understood as considering the compactiﬁed moduli space M I over A instead of2
MI/k(which is over k). The analogous problem, i.e., to construct a surjection fromthe rational d-th K-group of a compactiﬁcation of the moduli over A, remains open
for d ≥ 3.
In Section 8.1, we recall the deﬁnitions of the Bruhat-Tits building associatedto PGL2 of a non Archimedean local ﬁeld, and harmonic cochains. In Section 8.2,
we recall the deﬁnition of the regulator map, which is a homomorphism from theK2 of the Drinfeld modular curve to the space of Z-valued harmonic cochains. The
details on the regulator map will be used in the proof of Lemmas 9.2 and 9.4.
8.1. Bruhat-Tits building and harmonic cochains.
8.1.1. Let BT denote the Bruhat-Tits building for PGL2 (k∞ ). It is a simplicial
complex of dimension one. The set of vertices BT 0 (resp. of oriented edges BT 1 )××
is canonically isomorphic to GL2 (k∞ )/k∞
GL2 (O∞ ) (resp. GL2 (k∞ )/k∞
I whereI ⊂ GL2 (O∞ ) is the Iwahori subgroup).
For v ∈ BT 1 , let s(v) ∈ BT 0 denote the source of the oriented edge v. For
an abelian group B, a P
B-valued harmonic cochain on BT is an element f ∈Map(BT 1 , B) such that s(v)=e f (v) = 0 holds for all v ∈ BT 0 and f (ev1 ,v2 ) =−f (ev2 ,v1 ) where ev1 ,v2 means the oriented edge with source v1 and target v2 .
`8.1.2. Let I ⊂ A be a nonzero ideal. Let BT I = GL2 (A∞ )/KI ×BT = γ∈GL2 (A∞ )/KI BT(recall KI from Section 4.3.2). We let HI denote `the Z-module of Z-valued harmonic cochains on BT I (i.e., an element of Map( γ∈GL2 (A∞ )/KI BT 1 , Z) that is
a harmonic cochain when restricted to a component) that are GL2 (k)-invariant.
We regard HI as a submodule of Map(GL2 (k)\GL2 (A)/KI I, Z) via the canonical×
I. We let H0,I ⊂ HI denote the submodule ofisomorphism BT 1 ∼= GL2 (k∞ )/k∞elements which have compact support.
We let H = limI HI and H0,I = limI H0,I . We regard them as subsets H0 ⊂−→−→×
I, Z).
H ⊂ Map(GL2 (k)\GL2 (A)/k∞
8.1.3. For an abelian group B, we let HB and H0,B denote H ⊗Z B and H0 ⊗Z B
respectively. There is an
R analogue of Petersson inner product h , i : H0,Q ×HQ → Q
f (g)f2 (g)dg, where dg denotes the Haardeﬁned as hf1 , f2 i = GL2 (k)k∞×
Q \GL2 (A) 1
measure of GL2 (A) with vol( ℘ GL2 (A℘ )) = 1. Since the restriction of h , i to
H0,I,Q ×H0,I,Q is non-degenerate, there exists a unique homomorphism PI : HI,Q →H0,I,Q such that hf1 , f2 i = hf1 , PI f2 i for all f1 ∈ H0,I,Q , f2 ∈ HI,Q . Passing to the
inductive limit we obtain a GL2 (A)-equivariant homomorphism P : HQ → H0,Q .
8.2. Regulator. Let BT denote the Bruhat-Tits building for PGL2 (k∞ ). Let
I ⊂ A be a non-zero ideal such that Spec (A/I) contains two or more closed points.
In [Ko-Ya, p.22, Section 7.3], the authors deﬁned the regulator map
2
regI : K2 (MI/k) → HI .

Let us recall the construction.
Let X = X2 be the Drinfeld upper half plane (the Drinfeld symmetric space ofdimension one) over k∞ . The rigid analytic uniformization of MI2 (see Section 4.3.4(4.2)) gives a canonical morphisma2
X → (MI/k)an∞γ∈GL2 (A∞ )/KI40

SATOSHI KONDO AND SEIDAI YASUDA

2
of rigid analytic spaces. For γ ∈ GL2 (A∞ ), let vγ : X → (MI/k)an denote its∞
γKI -component.
There is the specialization map: sp : X → BT or the reduction map [DeS, PartII, 6.1, p. 157] (we note that this specialization map is diﬀerent from the map sp
introduced in Section 4.3). Let e be an oriented edge (including the endpoints)
in the Bruhat-Tits building BT . Then sp−1 (e) is isomorphic to Spm Be for some
aﬃnoid algebra Be . Let Ye denote the special ﬁber of the formal model of Spm Be .
It is known that Ye has two irreducible components Cs and Ct (corresponding to
the source and the target of the edge e) intersecting at a κ(℘)-rational point. Each
irreducible component is isomporphic to an open subset of the projective line over2
)an denote the restriction of vγ to sp−1 (e).
κ(℘). Let vγ,e : Spm Be → (MI/k∞
Associated to vγ,e , by the functoriality of rigid analytic functions for a morphisms
2) → Be .
of rigid analytic spaces, there is a canonical ring homomorphism O(MI/k∞ConsiderYY∂
2
G1 (Ye )K2 (Be ) −→(8.1)
K2 (MI/k
)→
γ,e

γ,e

where the ﬁrst map is the pullback map induced by the canonical ring homomorphism above, the second is the boundary map in the localization sequence of the pair(Spec Be , Spec Be /(π∞ )) (π∞ is a uniformizer of k∞ ). Let p denote the intersectionof the two irreducible components of Ye . We have G1 (Ye ) → G1 (Cs \ {p}) → G0 (p)where the ﬁrst map is the restriction map and the second map is the boundary map
in the localization sequence.
2
Let x ∈ K2 (MI/k). For γ ∈ GL2 (A∞ )/KI and v ∈ BT 1 . We deﬁne regI (x)((γ, v))
to be the image of x under the composition of the maps above to G0 (p) ∼= Z.
8.3. Let us state the results from our other paper [Ko-Ya].
For each non-zero ideal I ⊂ A such that Spec (A/I) contains two or more closed222
points, let WI = Image[O(MI/k)× ⊗ O(MI/k)× → K2 (MI/k)]. We put W =limI WI .
−→Lemma 8.1. Let the notations be as above. The composite
reg

reg0Q : WQ −−→ HQ −→ H0,QP

is surjective.
Proof. It follows from [Ko-Ya, p.23, Theorem 9.1].

¤

9. On the compactification
The aim of this section is to prove Proposition 9.1. This is the only result to be
used in other sections, and will be used only in Section 11.1.
Lemma 8.1 says that there are certain nontrivial elements in the K2 of the(open) Drinfeld modular curves. In this section, we construct elements in the K2 ofthe compactifcation of the Drinfeld modular curves, starting from those elements,
without modifying the property (namely the surjectivity of certain homomorphism)at the prime ∞.
The details of the procedure in the case of the moduli of elliptic curves are given
in [De-Wi]. The key ingredient there is the Drinfeld-Manin theorem on the ﬁnitenessof the cuspidal divisor group of Drinfeld modular curves. Using its Drinfeld modular

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

41

analogue proved by Gekeler ([Gek1]), we give the details for the Drinfeld modularcase in this section.
In Section 9.1, we give the statement of the main result of this section. The resultof Section 9.2 has a rather direct analogue in the elliptic modular case. Lemma 9.4
of Section 9.3 requires some computation regarding K-groups. The procedures in
the proof of Lemma 9.4 would be much clearer if we had K-theory for rigid analytic
spaces. As we do not, we do some diagram chasing.
9.1. For each prime ℘ of k, the homomorphism
2

c2,2

2

2

2
K2 (M I/k ) → K2 (M I/k℘ ) −−→ Hét
(M I/k℘ , Q` (2))
222gives rise to the homomorphism K2 (M lim,k ) → Hét
(M lim,k℘ , Q` (2)) which we denote by c2,2,℘ .
For each non-zero ideal I ⊂ A such that Spec (A/I) contains two or more closed222
points, let WI = Image[O(MI/k)× ⊗ O(MI/k)× → K2 (MI/k)]. We let W0,I denote22
the inverse image of WI under the homomorphism K2 (M I/k ) → K2 (MI/k). We
put W = limI WI and W0 = limI W0,I .
−→−→Proposition 9.1. Let ℘ be a prime of k. The composite
2c2,2,℘

2

2
W0 ⊗Z Q` → K2 (M lim,k ) ⊗Z Q` −−−→ Hét
(M lim,k℘ , Q` (2))

is surjective if ℘ = ∞. If ℘ 6= ∞, then it equals zero.
Proof. The case ℘ 6= ∞ follows from Proposition 7.6 using Proposition 3.1. The
rest of this section is devoted to the proof of the case ℘ = ∞.
9.2. Pulling back by the determinant morphism (deﬁned in Section 4.4), we may
12
regard elements of O(MI/k) as elements of O(MI/k).
12
Lemma 9.2. Let u1 ∈ O(MI/k)× and u2 ∈ O(MI/k)× . Then P ◦ regI ({u1 , u2 })
equals zero.

Proof. From the deﬁnition of regI (Section 8.2), we haveµ¶
ordCt(e) u1 ordCt(e) u2
regI ({u1 , u2 })(e) = det
ordCs(e) u1 ordCs(e) u2
where s(e) (resp. t(e)) is the source (resp. the target) of the oriented edge e,
and Cv is the component corresponding to the vertex v (see Section 8.2). As1
u1 ∈ O(MI/k)× , we see that there exists a function h on the set of vertices suchthat regI ({u1 , u2 })(e) = h(t(e)) − h(s(e)), i.e., it is a 1-coboundary.
For a harmonic cochain f ∈ H0,I with compact support, we have
X X
X X
X
f (e)[h(t(e))−h(s(e))] =f (e)h(t(e))−
f (e)h(s(e)).
hf, regI ({u1 , u2 })i =evt(e)=vvs(e)=vBoth quantities on the right hand side are zero by the harmonicity of f . Thisﬁnishes the proof.
¤
Let the notations be as above.

42

SATOSHI KONDO AND SEIDAI YASUDA

Lemma 9.3. The composite
reg

P

2
W0,Q → WQ → K2 (Mlim,k)Q −−→ HQ −→ H0,Qis surjective.
Proof. We apply the method of [De-Wi, Lemma 5.2, p. 265] in the case of elliptic
modular curves (called Bloch’s method there) to our Drinfeld modular context.
Let h0 ∈ H0,Q . By Lemma 8.1, one can ﬁnd an element κ in the image of the2
map WI,Q → K2 (MI/k)Q which maps to h0 via P ◦ regI for some I. We show that2

there exists an element in W0,I,Q ⊂ K2 (M I/k ) which maps to κ via the restriction22). This implies the statement of the lemma.
map K2 (M I/k ) → K2 (MI/k2

1
We note that every cusp of M I/k is an MI/k-valued point (see [Le, p.82, Propo1
sition 3.5]). We also recall that MI/k is the spectrum of a ﬁeld (see Lemma 4.2).
Consider the exact sequence
2

ρ

µ

2

2

ν

2
2
K2 (M I/k ) −→ K2 (MI/k)−→ K1 (M I/k \ MI/k)−
→ K1 (M I/k ).
2

The last term K1 (M I/k ) tensored with Q contains a direct summand isomorphic2

to (O(M I/k )× )Q (see, for example, [Bl1, p.267-269]). Using Lemma 6.1(2), weM2
see that (O(M )× )Q ∼O(M 1 )× == (O(M 1 )× )Q . An element (fc )c ∈
I/k

I/k

I/k

c

Q

2

2
K1 (M I/k \ MI/k
) is sent via ν to (the class of) c fc in the summand. Here c runs2
over the set of cusps in each case. Let κ ∈ K2 (MI/k), and let (fc )c be its image inQ
2
2
K1 (M I/k \ MI/k ). By the exactness, we have c fc = 1.
The Drinfeld modular analogue of the theorem of Drinfeld and Manin is provedby Gekeler ([Gek1, 1.2 Theorem, p. 359]). (Gekeler proves that the cuspidal divisor
class group of the Drinfeld modular curve associated to a congruence subgroup Γ,
denoted M Γ there, which is deﬁned over some ﬁnite extension of the base ﬁeld, is
ﬁnite. To apply it to our case, use that there exists a ﬁnite ﬁeld extension L ⊃ k such2
that there exists a ﬁnite ﬂat map h : M I/L → M Γ which sends cusps to cusps. Thenas h∗ h∗ = deg h on the G-group we obtain N below.) We can take an integer N
large enough so that N annihilates the cuspidal subgroup. Let us ﬁx a cusp, say c0 ,2
and consider an element gc ∈ O(MI/k)× whose divisor is N ([c] − [c0 ]) for each cusp
P
2×
1)× . Then the image of u under thec. We put u = c gc ⊗fc ∈ O(MI/k ) ⊗O(MI/k
2

2
1
2
2
map O(MI/k)× ⊗O(MI/k)× → K2 (MI/k) → K1 (M I/k \MI/k) equals N µ(κ). Thus222
for each κ ∈ K2 (MI/k), one can ﬁnd an element κ in Image[K2 (M I/k ) → K2 (MI/k
)]
2

2
1
such that κ − N κ belongs to Image[O(MI/k)× ⊗ O(MI/k)× → K2 (M I/k )]. Using
Lemma 9.2, we obtain the claim.
¤

9.3.
Lemma 9.4. The kernel of regI contains the kernel of the composite
c2,2

2
2
2
2
K2 (MI/k) → K2 (MI/k) −−→ Hét
(MI/k, Q` (2)).
∞∞

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

43

Proof. By the deﬁnition of regI the kernel of regI contains the kernel of the composite(9.1)

2
K2 (MI/k)→

Y∂

K2 (Be ) −
→

γ,e

Y

G1 (Ye ) →γ,eY

G1 (Ye )/G1 (Ye )tors

γ,ewhere the ﬁrst two maps are as in (8.1) and the third is the canonical surjection.
2
Let K, Le denote the total quotient ring of O(MI/k), O(Ye ), respectively. Let∞
Frac(Be ) denote the ﬁeld of fractions of Be . Consider the following commutative
diagram:(1)2
K2 (MI/k) −−−−→ ∞(2)yK2 (K)

K2 (Be )


(3)y

(4)

∂

−−−−→ G1 (Ye )/G1 (Ye )tors


(5)y
∂

−−−−→ K2 (Frac(Be )) −−−−→ K1 (Le )/K1 (Le )tors

where the map (1) is the map induced by vγ,e ; the maps (2) (3) (4) are thoseinduced by the canonical homomorphisms of rings; the top ∂ is as in (8.1); thebottom ∂ is the norm map of Milnor K-theory with respect to the valuation givenby the uniformizer of k∞ ; the vertical map (5) is induced by the pullback map to
the generic points of the irreducible components.
We note that the homomorphism (5) is injective since Ye is a normal crossing
curve over a ﬁnite ﬁeld. Hence the kernel of regI contains the kernel of the composite
2
K2 (MI/k) → K2 (K) →

Y
γ,e

∂

K2 (Frac(Be )) −→Y

K1 (Le )/K1 (Le )tors .

γ,eFor an abelian group M , let M ∧ = (limn M/ln ) ⊗Z Q. The following diagram is←−commutative:Q
K2 (K) −−−−→
γ,e G1 (Le )/G1 (Le )tors




(1)y
y
Q
K2 (K)∧ −−−−→ ( γ,e G1 (Le )/G1 (Le )tors )∧ .
Here the vertical maps are the canonical maps; the map at the top is the map in the
diagram above; the map at the bottom is the map induced by the map at the top.
Since Le is the product of two copies of the ﬁeld of rational functions of one variable
over a ﬁnite ﬁeld, G1 (Le )/G1 (Le )tors is a free abelian group. Hence the map (1) is
injective. Therefore, the kernel of regI contains the kernel of the composite(9.2)

2
K2 (MI/k) → K2 (K) → K2 (K)∧ .

We note that K is the product of a ﬁnite number of ﬁelds. By the theoremof Merkurjev and Suslin ([Me-Su, (11.5), Theorem, p. 328]), the symbol map
22
K2 (K) → Hét(Spec K, Q` (2)) gives an isomorphism K2 (K)∧ ∼(Spec K, Q` (2)).
= Hét
The following diagram, where the maps (1) and (3) are those induced by the canonical inclusions of rings, the map (2) is the canonical map, and the map (4) is the44

SATOSHI KONDO AND SEIDAI YASUDA

pullback, is commutative:(1)

2
K2 (MI/k) −−−−→(3)yK2 (K)(2)

K2 (K)∧

∼
y=

−−−−→

(4)

2
2
2
) −−−−→ H 2 (MI/k, Q` (2)) −−−−→ Hét(Spec K, Q` (2)).
K2 (MI/k
∞
∞
c2,2

2
)→
Hence the kernel of (9.2) contains the kernel of the composite map K2 (MI/k22
2¤
K2 (MI/k∞ ) → H (MI/k∞ , Q` (2)). This completes the proof of Lemma 9.4.

9.4. Proof of Proposition 9.1.
Proof of Proposition 9.1. Let W 0 denote the image of2

c2,2,∞

2

2
W0,Q` → K2 (M lim,k )Q` −−−−→ Hét
(M lim,k∞ , Q` (2)).

Using Lemma 9.4 and 9.3, we see that H0,Q` is a quotient of W 0 .
We have ([vdP-Re, Theorem 2.13, p.199]) a canonical isomorphism H0,Q` ∼=HomGL2 (k∞ ) (St, Acusp ) of GL2 (A∞ )-modules, where Acusp denotes the space of Q` valued cusp forms on GL2 (k)\GL2 (A), and St denotes the Steinberg representationof GL2 (k∞ ). Let V` be a two dimensional Q` -representation of Gal(k ∞ /k∞ ) which
admits a non-split exact sequence
0 → Q` → V` → Q` (−1) → 0.
Such V` is unique up to an isomorphism. By the fundamental theorem in [Dr1, §11,
p. 591], we have an isomorphism
2
2
2
1
Hét
(M lim,k∞ , Q` (2)) ∼H 1 (k∞ , Hét
(M I/k∞ , Q` (2)))
= lim
−→1 I
∼
= H (k∞ , V` ) ⊗Q` HomGL2 (k∞ ) (St, Acusp )22of GL2 (A∞ )-modules. Since H 1 (k∞ , V` ) is one-dimensional, Hét
(M lim,k∞ , Q` (2)) is∞
isomorphic to H0,Q` as a GL2 (A )-module. On the other hand, H0,Q` is a quotient22of the GL2 (A∞ )-submodule W 0 of Hét
(M lim,k∞ , Q` (2)). Using the multiplicity one22(M lim,k∞ , Q` (2)), since H0,Q`
theorem, we conclude that the module W 0 equals Hét
is an admissible GL2 (A∞ )-module. This completes the proof of Proposition 9.1. ¤10. Modular parametrization
The aim of this section is to recall the modular parametrization of an elliptic
curve with a split multiplicative reduction. The reader is referred to [Ge-Re] forthe details including some historical account. Let E be an elliptic curve over a
global ﬁeld k of positive characteristic, and suppose that it has split multiplicative
reduction at a prime ∞. Then a modular parametrization is a non-constant morphism from a Drinfeld modular curve (whose prime at inﬁnity is the prime ∞) tothe ellitpic curve. For (the conditions on) the existence and the construction, thereader is referred to Section 8 of [Ge-Re].
In Section 9, we have considered elements in the rational K2 of the compactiﬁed
Drinfeld modular curves (namely, the image of the ﬁrst map in Proposition 9.1).
We use the pushforward map by the modular parametrization to obtain elementsin the rational K2 of an elliptic curve over k. This will be done in Section 11.1.

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

45

Lemma 10.1 describes the map which is induced by the parametrization on theétale cohomology group of our interest. The surjectivity in question on the cohomology reduces to the surjectivity of the morphism of the Jacobians, which isimmediate since the morphism is non-constant and the target is an elliptic curve.
10.1. Let E be an elliptic curve over a global ﬁeld k of positive characteristic,
and let C be the projective smooth curve over a ﬁnite ﬁeld whose function ﬁeld
is k. We ﬁx a prime ∞ of k at which E has split multiplicative reduction. Let
A = H 0 (C \ {∞}, OC ). We use the notations in Section 4 and Section 5. We also
ﬁx a separable closure k ∞ of k∞ .
Let I ( A be a non-zero ideal such that Spec (A/I) contains two or more closed
2dpoints. For an A-algebra L, we let MI/L
= MId ×Spec A Spec L and M I/L =2

M I ×Spec A Spec L. For a k-algebra L0 , we let EL0 = E ×Spec k Spec L0 .
2
Recall that M I denotes the compactiﬁcation of the A-scheme MI2 as constructedby Drinfeld (see the proof of 6.1 where it ﬁrst appeared). For a ﬁeld L ⊃ k, let us2222
denote by Mlim,L(resp. M lim,L ) the projective limit limI (MI/L) (resp. limI (M I/L ))←−←−where I runs over the ideals such that Spec (A/I) contains two or more closed
points. These projective limits exist in the category of schemes over Spec L since
all the transition maps are ﬁnite.
Let IE denote the ﬁnite part of the conductor, regarded also as an ideal of A.
We use the non-constant morphism prE : J0 (IE ) → E of [Ge-Re, 8.3.10, p.82] (seeloc. cit. for the notation J0 (IE ) and M02 (IE ) where they use n for IE ). We use
22
it in the form ϕE : M IE /k → M 0 (IE ) → J0 (IE ) → E where the ﬁrst map is the
canonical quotient map, the second map is the canonical inclusion into the Jacobian,and the third map is prE above. For J ⊂ IE , let ϕE,J denote the composite
22
2
2
M J/k → M I/k → E. For a ﬁeld L ⊃ k, we let ϕE,J,L : M J/L → M I/L → EL
denote the base change of ϕE,J to L.
We ﬁx a prime number ` diﬀerent from p. For each prime ℘ of k, we put
2
2
2
2
(MI/k
, Q` (2)),
Hét(Mlim,k, Q` (2)) = limI Hét
℘
℘
−
→
2
2
2
2
Hét (M lim,k℘ , Q` (2)) = limI Hét (M I/k℘ , Q` (2)).
−→
22where the transition map for the ﬁrst line is the pullback map ϕ∗J,J 0 : Hét
(MJ/k
, Q` (2)) →
℘
2
2
0
Hét
(MJ20 /k℘ , Q` (2)) where ϕJ,J 0 : MJ20 /k℘ → MJ/k
for
J
⊂
J
is
the
canonical
level℘lowering morphism. The transition map for the second line is deﬁned in a similarmanner.
Consider the homomorphisms

12
2
2· ϕE,J,k℘ ∗ : Hét
(M J/k℘ , Q` (2)) → Hét
(Ek℘ , Q` (2))
deg(ϕE,J,k℘ )where J ⊂ IE is an ideal such that Spec (A/J) contains two or more closed
points. Let J 0 ⊂ J ⊂ IE be two ideals with the same condition. Then since22
ϕJ,J 0 ∗ ϕ∗J,J 0 = deg ϕJ,J 0 , we obtain a homomorphism ϕ∗,℘,ét : Hét
(M lim,k℘ , Q` (2)) →1
2
Hét(Ek℘ , Q` (2)) as the limit of the homomorphisms deg(ϕE,J,k) · ϕE,J,k℘ ∗ .
℘Lemma 10.1. The homomorphism ϕ∗,℘,ét is surjective.

46

SATOSHI KONDO AND SEIDAI YASUDA

Proof. Let us consider the morphism ϕE,I for a suﬃciently small ideal I. Take aseparable closure k ℘ of k℘ . We have a commutative diagram
2

2

2
1
Hét
(M I/k℘ , Q` (2)) −−−−→ H 1 (k℘ , Hét
(M I/k℘ , Q` (2)))

ϕE,I,k℘ 
y
∗y
2
Hét
(Ek℘ , Q` (2))

1−−−−→ H 1 (k℘ , Hét
(Ek℘ , Q` (2))).

Here we denote by H ∗ (k℘ , −) the continuous Galois cohomology, and the horizontal arrows are the maps which appear in the Leray spectral sequence, tak2
2
2
(Ek℘ , Q` (2))) =ing into account that H 0 (k℘ , Hét
(M I/k℘ , Q` (2))) = H 0 (k℘ , Hét0
H (k℘ , Q` (1)) = 0; the right vertical arrow is the map induced by ϕE,I∗ . Since200
H 2 (k℘ , Hét
(M I/k℘ , Q` (2))) = H 2 (k℘ , Hét
(Ek℘ , Q` (2))) = 0 by the local Tate duality, the horizontal arrows are isomorphisms. Since ϕE,I is non-constant, thehomomorphism
2

1
1
Hét
(M I/k℘ , Q` (2)) → Hét
(Ek℘ , Q` (2))

of Gal(k ℘ /k℘ )-modules is split surjective. Hence ϕE,I,k℘ ∗ is surjective.

¤

11. Proof of Theorem 1.1
Let C, E, E, S be as in the statement of Theorem 1.1. We give a proof of
Theorem 1.1 and Corollaries 1.2 and 1.3 in this section. In Section 11.1, we use
results from previous sections (Proposition 9.1, and Lemma 10.1) to construct anelement in K2 (E)Q for each place of split multiplicative reduction. We keep track of
the properties (i.e., under the boundary maps) of the elements using Proposition 3.1
among other things. Note that this construction does not have an analogue in thenumber ﬁeld analogue, since there is a clear distinction between the inﬁnity prime
and the ﬁnite prime in the number ﬁeld case. Section 11.3 is devoted to the proofsof the statements in the introduction.
11.1. Proof of Theorem 1.1 when there is a section. In this section, we
prove Theorem 1.1 assuming that E → C has a section.
When S = ∅, the target space is zero and there is nothing to prove. Assume
S 6= ∅ and take s ∈ S. We use the notations in Section 9 for ∞ = s.
For a k-algebra L, we let Ek = E ×Spec k Spec L Using the computation done in
2
1
the proof of Lemma 10.1, we know that Hét
(Ek∞ , Q` (2)) = H 1 (k∞ , Hét
(Ek̄∞ , Q` (2))).
1
Since E has split multiplicative reduction at ∞, we have Hét (Ek̄∞ , Q` (2)) = V`2
where V` is as in Section 9.4, and hence Hét
(Ek∞ , Q` (2)) is one dimensional.
Using Proposition 3.1 and that the dimension of G1 (E℘ ) is less than or equal
to 1 by Lemma 2.2, we see that the statement of Theorem 1.1 is equivalent to thesurjectivity of⊕℘ c2,2,℘ M
2Hét(Ek℘ , Q` (2))
K2 (E)Q` −−−−−→℘

where ℘ runs over all primes of k.
Combining Proposition 9.1 with Lemma 10.1, we see that the composite
2c2,2,∞

2

2
W0 ⊗Z Q` → K2 (M lim,k )Q` −−−−→ Hét
(M lim,k∞ , Q` (2))ϕ∗,∞,ét2−−−−−→ Hét (Ek∞ , Q` (2))

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE

47

2
is surjective. We let κ ∈ W0 denote a preimage of a nonzero element in Hét
(Ek∞ , Q` (2)).
Let us consider the morphism ϕE,I for a suﬃciently small I. The homomorphism

1
2· ϕE,I ∗ : K2 (M I/k ) → K2 (E)Q
deg(ϕE,I )2

gives rise to the homomorphism ϕ∗,K : K2 (M lim,k ) → K2 (E)Q . We put κs =ϕ∗,K (κ) ∈ K2 (E)Q .
Lemma 11.1. For s, s0 ∈ S, we have ∂s0 (κs ) = 0 if and only if s 6= s0 .
Proof. For each s0 ∈ S, we have a commutative diagram
c2,2,s02

K2 (M lim,k ) −−−−→

ϕ∗,K
y
K2 (E)

2

2(M lim,ks0 , Q` (2))
Hét
ϕ∗,s0 ,étyc2,2,s02−−−−→ Hét
(E ×Spec k Spec ks0 , Q` (2)).

If s 6= s0 , then by construction of κs , we have c2,2,s0 (κs ) = ϕ∗,s0 ,ét (c2,2,s0 (κ)) = 0
using Proposition 9.1. By Proposition 3.1, the claim holds in this case.
Suppose s = s0 . Then c2,2,s (κs ) = ϕ∗,s0 ,ét (c2,2,s (κ)) 6= 0 by construction. Using
Proposition 3.1, we obtain the claim.
¤
Consider the subspace of K2 (E)Q generated by {κs |s ∈ S}. Using Lemma 11.1
one sees that it maps surjectively onto the right hand side. This completes theproof of Theorem 1.1 when there is a section.
¤
11.2. Proof of Theorem 1.1. Let E → C be as in the setup of Theorem 1.1. We
do not assume the existence of a section to E → C. Let D be a projective smooth
curve over a ﬁnite ﬁeld with a ﬁnite ﬂat morphism ϕ : D → C such that the basechange ED = E ×C D → D has a section. Let E 0 → ED denote a desingularization.
The argument above (the proof of Theorem 1.1) applies to E 0 → D.
Let s ∈ C and t ∈ ϕ−1 (s).
Lemma 11.2. We havedimQ G1 (Es )Q ≤ dimQ G1 (Et0 )Q ≤ 1.
Proof. From Proposition 2.1, we have dimQ G1 (Eks ) = dimQ` H 2 (Eks , Q` (2)) anddimQ G1 (Ek0 0 ) = dimQ` H 2 (Ek0 0 , Q` (2)) where we write k 0 for the function ﬁeld of Dtt
and E 0 = E ×Spec k Spec k 0 . We also write Et0 = E 0 ×D t and Ek0 0 = E 0 ×Spec k0 Spec kt0twhere kt0 is the local ﬁeld at t.
The canonical morphism ϕts : Ek0 0 → Es induced from ϕ is ﬁnite ﬂat. Sincet
ϕts∗ ϕ∗ts = deg ϕts on H 2 (Eks , Q` (2)), we obtain the ﬁrst inequality. The second
inequality is the content of Lemma 2.2.
¤
Let s ∈ C be a closed point where dimQ G1 (Es )Q = 1. Choose an element t ∈ϕ−1 (s). From the argument in Section 11.1, it follows that there exists an element
κt ∈ K2 (E 0 )Q such that ∂t0 κt = 0 for t0 6= t and ∂t κt 6= 0. Using Proposition 3.1,
we see that c2,2,t0 κt = 0 for t0 6= t and c2,2,t κt 6= 0.

48

SATOSHI KONDO AND SEIDAI YASUDA

Let v ∈ C and consider the following commutative diagram:M⊕w c2,2,w
0
K2 (E 0 )Q −−−−−−−→
H 2 (Ew, Q` (2))
w∈ϕ−1 (v)ϕ∗ yK2 (E)Q

c2,2,v

−−−−→


P ϕ
y w wv∗

H 2 (Ev , Q` (2)),

where ϕwv is the restriction of the morphism ϕ.
We then see that c2,2,s0 κs = 0 for s0 6= s and c2,2,s κs = 0, since ϕts∗ is an
isomporphism if dimQ G1 (Es )Q = 1. Using Proposition 3.1, we have the desired
properties of the element κs as in the proof of Theorem 1.1 above. The rest of theproof is the same as in Section 11.1. This completes the proof.
¤
11.3. Proofs.
Proof of Corollary 1.2. Let C be the projective smooth curve over a ﬁnite ﬁeld
whose function ﬁeld is k, and let E be the ﬂat proper regular minimal model of E
over C. The claim follows from Theorem 1.1 for E → C, after we take (2.1) into
account.
¤
Proof of Corollary 1.3. Let C be the projective smooth curve whose function ﬁeld
is k. We choose a closed point c and let A = H 0 (C \ {c}, OC ) denote the coordinate
ring of the aﬃne smooth curve C \ {c}. It is a Dedekind domain.
Let {℘i }1≤i≤n be a set of nonzero prime ideals of A. Take a nonzero elementa ∈ ∩i ℘i . Consider the Weierstrass equation y 2 + xy = x3 + a. Note that modulo℘i , it becomes the singular curve deﬁned by y 2 + xy = x3 . It has a node at (0, 0)with tangent lines y = 0 and y + x = 0 (see [Si, p.44, Deﬁnition]). Hence theelliptic curve over the function ﬁeld of C deﬁned by the equation above has split
multiplicative reduction at ℘i for each i. Now apply Corollary 1.2 to deduce theclaim.
¤
References[Ba-Mi-Se] Bass, H., Milnor, J. W., Serre J.-P.: Solution of the congruence subgroup problem
for SLn (n ≥ 3) and Sp2n (n ≥ 2). Publ. Math., Inst. Hautes Étud. Sci. 33, 59–137(1967)[Be]
Beilinson, A. A.: Higher regulators and values of L-functions. J. Sov. Math. 30,
2036–2071 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24,
181–238 (1984)[Bl1]
Bloch, S.: Algebraic cycles and higher K-theory. Adv. Math. 61, no. 3, 267–304 (1986)[Bl-Gr]Bloch, S., Grayson, D.: K2 and L-functions of elliptic curves: computer calculations.
In: Bloch, S. J., Dennnis, R. K., Friedlander, E. M., Stein, M. R. (ed.) Applicationsof algebraic K-theory to algebraic geometry and number theory, Boulder/Colo. 1983,
Contemp. Math. 55, Part I, 79–88. American Mathematical Society, Providence, RI(1986)[Bl-St]Blum, A., Stuhler, U.: Drinfeld modules and elliptic sheaves. In: Narasimhan, M. S.
(ed.) Vector bundles on curves—new directions, Cetraro (Cosenza), Italy, June 19-27,
1995, Lect. Notes Math. 1649, 110–193. Springer, Berlin (1997)[Bo]
Boyer, P.: Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands local. Invent. Math. 138, no. 3 573–629 (1999)[Co]
Cogdell, J. W.; Kim, H. H.;Murty, M. R., Lectures on automorphic L-functions. Fields
Institute Monographs 20. (2004)

ON THE RATIONAL K2 OF AN ELLIPTIC CURVE[De-Wi]

[DeS]
[Dr1]
[Dr2]
[Dr3]
[Fu]

[Gek1][Gek2][Gek3][Gek4][Gek5][Ge-Re][Ge-Ja]

[Gen]
[Gi]
[Ha]
[vdHe][Ja-La][Ja-Sa][Ka-Ma][Ka-Sa][Ko-Ya][La-Mo][Le][Liu][Me-Su]49

Deninger, C., Wingberg, K.: On the Beı̆linson conjectures for elliptic curves with complex multiplication. In: Rapoport, M., Schappacher, N., Schneider, P. (ed.) Beı̆linson’sconjectures on special values of L-functions, Oberwolfach, FRG in April 1986, Perspect. Math. 4, 249–272. Academic Press, Boston, MA (1988)
De Shalit, E.: Residues on buildings and de Rham cohomology of p-adic symmetricdomains. Duke Math. J. 106, no. 1, 123–191 (2001)
Drinfeld, V. G.: Elliptic modules. Math. USSR, Sb. 23, 561–592 (1974); translationfrom Mat. Sb., n. Ser. 94(136), 594-627 (1974)
Drinfeld, V. G.: Elliptic modules. II. Mat. USSR Sb. 31, 159–170 (1977); translationfrom Mat. Sb., n. Ser. 102(144), no. 2, 182–194, 325 (1977)
Drinfeld, V. G.: Commutative subrings of some noncommutative rings. Funct. Anal.
Appl. 11, 9–12 (1977)
Fujiwara, K.: A proof of Absolute Purity Conjecture (after Gabber). In Usui, S. (ed.)
et al. Algebraic geometry 2000, Azumino. Proceedings of the symposium, Nagano,
Japan, July 20–30, 2000, Adv. Stud. Pure Math. 36, 153–183. Mathematical Societyof Japan, Tokyo (2002)
Gekeler, E.-U.: A note on the ﬁniteness of certain cuspidal divisor class groups. Isr.
J. Math. 118, 357–368 (2000)
Gekeler, E.-U.: On the cuspidal divisor class group of a Drinfeld modular curve. Doc.
Math., J. DMV 2, 351–374 (electronic) (1997)
Gekeler, E.-U.: On ﬁnite Drinfeld modules. J. Algebra 141, 187–203 (1991)
Gekeler, E.-U.: Über Drinfeldsche Modulkurven vom Hecke-Typ. Compos. Math. 57,
no. 2, 219–236 (1986)
Gekeler, E.-U.: Zur Arithmetik von Drinfeld-Moduln. Math. Ann. 262, no. 2, 167–182(1983)
Gekeler, E.-U., Reversat, M.: Jacobians of Drinfeld modular curves. J. Reine Angew.
Math. 476, 27–93 (1996)
Gelbart, S., Jacquet, H.: Forms of GL(2) from the analytic point of view. Automorphic
forms, representations and L-functions, Proc. Symp. Pure Math. Am. Math. Soc.,
Corvallis/Oregon 1977, Proc. Symp. Pure Math. 33, 1, 213–251 (1979)
Genestier, A.: Espaces symétriques de Drinfeld. Astérisque 234, 124 p. (1996)
Gillet, H.: Riemann-Roch theorems for higher algebraic K-theory. Adv. Math. 40,
203–289 (1981)
Hamahata, Y.: Tensor products of Drinfeld modules and v-adic representations. Manscr. Math. 79, No. 3–4, 307–327 (1993)
van der Heiden, G.-J.: Weil pairing for Drinfeld modules. Monatsh. Math. 143, no.
2, 115–143 (2004)
Jacquet, H., Langlands, R. P.: Automorphic forms on GL(2). Lect. Notes Math. 114,
Springer-Verlag, Berlin-Heidelberg-New York (1970)
Jannsen, U.; Saito, S., Kato homology of arithmetic schemes and higher class ﬁeld
theory over local ﬁelds. Doc. Math., J. DMV Extra Vol., 479-538 (2003)
Katz, N. M., Mazur, B.: Arithmetic moduli of elliptic curves. Ann. Math. Stud. 108,
Princeton University Press, Princeton, NJ (1985)
Kato, K., Saito, S.: Unramiﬁed class ﬁeld theory of arithmetic surfaces. Ann. Math.
(2) 118, 241–275 (1983)
Kondo, S., Yasuda, S.: Zeta elements in the K-theory of Drinfeld modular varieties,
submitted
Laumon, G., Moret-Bailly, L.: Champs algébriques. Ergeb. Math. Grenzgeb., 3. Folge
39, Springer, Berlin (2000)
Lehmkuhl. T.: Compactiﬁcation of the Drinfeld modular surfaces. Mem. Am. Math.
Soc. 921, 94 p. (2009)
Liu, Q.: Algebraic geometry and arithmetic curves. Oxf. Grad. Texts Math. 6, Oxford
University Press, Oxford (2002)
Merkurjev, A. S., Suslin, A. A.: K-cohomology of Severi-Brauer varieties and thenorm residue homomorphism. Math. USSR, Izv. 21, 307–340 (1983); translation from
Izv. Akad. Nauk SSSR, Ser. Mat. 46, no. 5, 1011-1046 (1982)

50

SATOSHI KONDO AND SEIDAI YASUDA[Mi]

Milnor, J. W., Introduction to algebraic K-theory. Annals of Mathematics Studies.
No.72. Princeton, N. J.: Princeton University Press and University of Tokyo Press.
(1971)[Qu]
Quillen, D.: Higher algebraic K-theory I. In: H. Bass (ed.) Algebraic K-theory I,
Battelle Institute Conference 1972, Lect. Notes Math. 341, 85–147. Springer-Verlag,
Berlin-Heidelberg-New York (1973)[RSS]
Rapoport, M. (ed.); Schappacher, N.(ed.); Schneider, P.(ed.) Beilinson’s conjectures
on special values of L-functions (Based on the meeting of the Arbeitsgemeinschaft
Geyer-Harder held in Oberwolfach, FRG in April 1986). Perspectives in Mathematics,
4. Boston, MA etc.: Academic Press, Inc. (1988)[Ro-Sc]Rolshausen, K., Schappacher, N.: On the second K-group of an elliptic curve. J. Reine
Angew. Math. 495, 61–77 (1998)[Sc-Sc]Schappacher, N., Scholl, A. J.: Beı̆linson’s theorem on modular curves. In: Rapoport,
M., Schappacher, N., Schneider, P. (ed.) Beı̆linson’s conjectures on special values of
L-functions, Oberwolfach, FRG in April 1986, Perspect. Math. 4, 273–304. Academic
Press, Boston, MA (1988)[Si]
Silverman, J. H., The arithmetic of elliptic curves. 2nd ed. Graduate Texts in Mathematics 106. New York, NY: Springer. (2009)[St]
Stuhler, U.: P -adic homogeneous spaces and moduli problems. Math. Z. 192, 491–540(1986)[Th]
Thomason, R.W., The Lichtenbaum-Quillen conjecture for K/`∗ [β −1 ]. Current trends
in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, No.1, 117-139(1982)[vdP-Re]van der Put, M.; Reversat, M.: Automorphic forms and Drinfeld’s reciprocity law.
Gekeler, E.-U. (ed.) et al., Proceedings of the workshop on Drinfeld modules, modular
schemes and applications, Alden-Biesen, Belgium, September 9–14, 1996. Singapore:
World Scientiﬁc. 188-223 (1997)[Yu]
Yu, J.-K.: Isogenies of Drinfeld modules over ﬁnite ﬁelds. J. Num. Th. 54, 161–171(1995)[EGAII]Éléments de géométrie algébrique. II. Étude globale élémentaire de quelques classes
de morphisms. Publ. Math., Inst. Hautes Étud. Sci. 8, 1–222 (1961)[EGAIII1] Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents.
I. Publ. Math., Inst. Hautes Étud. Sci. 11, 1–167 (1961)[SGA4-2] Seminaire de geometrie algebrique Du Bois-Marie 1963-1964. Theorie des topos etcohomologie etale des schemas (SGA 4). Un seminaire dirige par M. Artin, A.
Grothendieck, J. L. Verdier. Avec la collaboration de N. Bourbaki , P. Deligne, B.
Saint-Donat. Tome 2. Exposes V a VIII. Lect. Notes in Math. 270 Berlin-HeidelbergNew York: Springer-Verlag (1972)[SGA4-3] Théorie des topos et cohomologie étale des schémas. Tome 3. Séminaire de géométrie
algébrique du Bois-Marie 1963-64 (SGA 4), Lect. Notes Math. 305, Springer-Verlag,
Berlin-Heidelberg-New York (1973)

Satoshi Kondo
Institute for the Physics and Mathematics of the Universe
University of Tokyoe-mail:satoshi.kondo@ipmu.jp

Seidai Yasuda
Research Institute for Mathematical Sciences
Kyoto University
e-mail:yasuda@kurims.kyoto-u.ac.jp