ON THE RATIONAL K2 OF A CURVE OF GL2 TYPE OVER A
GLOBAL FIELD OF POSITIVE CHARACTERISTICMASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

Abstract. Let k be the function field of a curve C over a finite field. Let X be aproper smooth curve over k. Then there exists a regular model X which is proper and
1 (X ×flat over C. Assume that the Q` -coefficient Gal(ksep /k)-representation HétSpec kSpec ksep , Q` ) is a direct sum of 2 dimensional irreducible representations.
∂⊗Q

We prove that the boundary map K2 (X) ⊗Z Q −−−→ G1 (X \ X) ⊗Z Q in the localization sequence of G-theory tensored with Q is surjective.
Examples satisfying the assumption above are elliptic curves, Drinfeld modular curves,and the moduli of D-elliptic sheaves of rank 2. Given an integer n, we also constructa genus two curve that satisfies the condition above and for which the dimension of
G1 (X \ X) ⊗Z Q is greater than n.

1. Introduction
We state our main result (Theorem 1.1 and Corollary 1.2) and give an outline of proofin this section.
1.1. Let Fq be a finite field of q elements of characteristic p. Let C be a projective smooth(geometrically connected) curve over Fq and k be the function field of C. Consider thefollowing cartesian diagram

/X
X
g




 η
Spec k f/Cwhere η is the generic point, f is proper flat, g is proper smooth, and X is regular. We
remark here that for a given proper smooth curve X over k, there exists an X as above by[22, p.456, Section 10, Proposition 1.8].
Let C0 denote the set of closed points of C. We regard an element in C0 as a place ofthe global field k and vice versa. We write (−)Q = (−) ⊗Z Q. The localization sequence
tensored by Q is the exact sequence
⊕∂℘ M
(1.1)
· · · → K2 (X )Q → K2 (X)Q →G1 (Xκ(℘) )Q → K1 (X )Q → · · · ,℘∈C0where κ(℘) is the residue field at ℘ and Xκ(℘) = X ×C Spec κ(℘). In this article, we
consider the following condition (GL2 ) on a proper smooth k-scheme Z. Let us write GL =Gal(Lsep /L) for the absolute Galois group of a field L, and ZM = Z ×Spec k Spec M for thebase change, where M is a k-algebra. Let ` be a prime different from p. The condition is
the following.
(GL2 )

1
The Gk -representation Hét
(Zksep , Q` ) is a direct sum of

2 dimensional irreducible representations.
Our main theorem is as follows.
Theorem 1.1. Let X be a proper smooth curve over k. Suppose that X satisfies Condition (GL2 ) (with Z = X). Let X be as above. Then the boundary map⊕∂℘ M
K2 (X)Q →G1 (Xκ(℘) )Q℘∈C0

is surjective.

1

2

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

This result is motivated by Parshin’s conjecture. Recall that Parshin’s conjecture saysthat Ki (Z) ⊗ Q = 0 for all i ≥ 1 when Z is a projective smooth scheme over Fq . With
our assumptions on X , the validity of the conjecture then implies that the boundary map⊕℘ ∂℘ in Theorem 1.1 is an isomorphism in view of the exact sequence (1.1).
We have the following corollary (see Corollary 6.1 for the precise statement).
Corollary 1.2. The surjectivity statement of Theorem 1.1 holds when X is(1) an elliptic curve,(2) a Drinfeld modular curve,(3) the moduli of D-elliptic sheaves of rank 2, or(4) one of the genus two curves constructed in Section 7.
In the four cases above, the curve X satisfies Condition (GL2 ). Hence this corollaryfollows from Theorem 1.1. In the paper [18], the theorem for the case where X is an elliptic
curve over k has been proved.
1.2. Let us give a detailed outline of proof of Theorem 1.1 below. First, we translate thetheorem into a statement about étale Chern class maps.
Proposition 1.3. Let the notation be as in Theorem 1.1. Suppose that Condition (GL2 )holds. Then the composite map
Q
2
2
HM(X, Q(2)) ⊗Q Q` → ℘∈C0 HM(Xk℘ , Q(2)) ⊗Q Q`
Q
2(Xk℘ , Q` (2)),→ ℘∈C0 Hétwhere the first arrow is the product of pullback maps and the second arrow is the product ofétale Chern class maps, is surjective.
To prove that Proposition 1.3 implies Theorem 1.1, we will use lemmas in the paper [18](see Section 5.3 for details).
We remark that we can compute the dimension of target group explicitly (see Section 6)2
and see that the factor Hét
(Xk℘ , Q` (2)) is zero for almost all ℘. It follows that the productof the étale cohomology groups is actually a (finite) direct sum.
1.3. To prove Proposition 1.3, we will show the following proposition.
Proposition 1.4. Suppose X = B is a simple abelian variety over k satisfying Condition (GL2 ). Then the map
Q
2,2 1
2,2 1
HM
(h (X)) ⊗Q Q` → ℘∈C0 HM(h (Xk℘ )) ⊗Q Q`
Q
2,2 1→ ℘∈C0 Hét (h (Xk℘ ))is surjective. Here we refer to Section 4 for the notation concerning Chow motives such asH∗2,2 and h1 . The first arrow is the pullback map and the second arrow is the étale Chernclass map.
To see that Proposition 1.4 implies Proposition 1.3, we use the formalism of Chow motives.
In the category of Chow motives (with rational equivalence and rational coefficients), we
have an isomorphism
h(X) ∼=

2M
i=0

hi (X) ∼= h0 (X) ⊕ h1 (JacX) ⊕ h2 (X) ∼= h0 (X) ⊕mM

h1 (Bi ) ⊕ h2 (X),i=1

where each Bi is a k-simple
Qm abelian variety satisfying Condition (GL2 ) such that JacX isisogenous (over k) to i=1 Bi (see Section 5.2). Then as étale Chern classes respect thedecomposition, and since the statement for the h0 and h2 is not an essential difficulty,
Proposition 1.3 is reduced to Proposition 1.4.
1
To prove Proposition 1.4, we proceed as follows. By Condition (GL2 ), we have Hét
(Bksep , Q` ) ∼=n⊕j=1 ρj where each ρj is an irreducible 2 dimensional representation of Gk with coefficientsin Q` . For a nonarchimedean local field L whose characteristic is prime to `, define Sp to be
the 2 dimensional representation of GL with coefficients in Q` which is the unique nontrivial
extension of Q` by Q` (−1). By abuse of notation we also write Sp for the base changeSp ⊗Q` Q` . We consider two cases separately.

THE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

3(1) There exists a pair (j, ℘) of an integer 1 ≤ j ≤ n and a place ℘ of k such that ρj |Gk℘
is isomorphic to Sp.
(2) The condition in (1) does not hold.
In Case (2), we show that the surjectivity statement becomes a trivial statement since, by
the computation of Galois cohomology in Section 3, we see that the target of the Chern class
map is zero (see Lemma 5.1). In Case (1), we show that B is isogenous to a simple factor of
the Jacobian of some Drinfeld modular curve such that ℘ is the fixed place at infinity. (Werefer to Section 2.1 for the relevant definitions and properties of Drinfeld modular curves.)
This is of course the function field analogue of Shimura’s theory [29]. We note that thecondition that there exists a place where the representation at the place ℘ is isomorphic to
Sp is the function field analogue of (or rather the Drinfeld modular analogue of) modular
forms being weight 2 (which is a condition at the place at infinity). As we did not find theconstruction in the literature, we give a detailed proof of this fact in Section 2. We notethat we do not follow the proof for the case of the moduli of elliptic curves, but we use freely
the Tate conjecture (a theorem of Zarhin and of Mori) to make the exposition simple. Withthe Tate conjecture at our use, the only other necessary ingredient is some Atkin-Lehnertheory over general global fields, but we use a theorem of Jacquet and Shalika for simplicity(see the proof of Lemma 2.3).
Thus we are reduced to proving Proposition 1.4 when B is a factor of the Jacobian of a
Drinfeld modular curve. The idea, which does not apply to the case of a number field, andwhich already appeared in the paper [18], is used at this point. Note that B may appear asa factor of a Drinfeld modular curve for EVERY ℘ satisfying the condition in (1), so that
we have as many “parametrizations” as such ℘. When B is an elliptic curve, this is reallythe modular parametrization as discussed in detail by Gekeler and Reversat [10]. We theninvoke the result of [18]. The setup is that of Proposition 1.3, and X is a Drinfeld modular
2
curve. The result in [18] says that there exists a certain subspace of HM(X, Q(2)) whichmaps surjectively by the Chern classes at infinity (or ℘ in this case) and maps to zero at all
the other places. By varying ℘, we obtain Proposition 1.4. This finishes the outline of theproof of Proposition 1.4 and hence of Theorem 1.1.
1.4. To see that the surjectivity statement is a nontrivial one, we will compute (Lemma 6.2)
the dimension of the target group explicitly in terms of the Gk -representation that appearsin Condition (GL2 ). More precisely, if the representation is the direct sum ⊕nj=1 ρj of 2
dimensional representations ρj , then the dimension of the target equals the number of thepairs (j, ℘) which appeared in the condition of (1).
In Corollary 1.2, we have four kinds of curves. For an elliptic curve, the condition on ℘above is equivalent to that the elliptic curve has split multiplicative reduction at ℘. For a
Drinfeld modular curve, the number of such pairs is bounded below by the genus since (j, ∞)always satisfy the condition of (1). Note that the genus grows as K becomes small. The
genus 2 curve in (4) is constructed from Brumer-Hashimoto’s family of genus 2 curves. We
construct explicit examples within this family for which the number of pairs (j, ℘) satisfying
the condition of (1) is large.
1.5. We give a remark concerning our title and Condition (GL2 ). Suppose a proper smooth
curve X satisfies Condition (GL2 ). Then the Jacobian of X is isogenous to a product ofsimple abelian varieties of GL2 -type if there exists a place ℘j such that (j, ℘j ) satisfies thecondition of (1) for each j. This can be seen using the procedure given in Section 5.1. When
there does not exist such ℘j ’s, we do not know if this holds true.
1.6. The paper is organized as follows. In Section 2, given an automorphic representation
π of GL2 (Ak ) such that π∞ ∼= St at some place ∞ of k (see Section 2 for the notation), weconstruct a k-simple abelian variety Aπ whose Tate module is related to π. Section 3 is devoted to the computation of Galois cohomology. We use the classification of representationsof GL2 (F ) (see Lemma 3.4) for a nonarchimedean local field F . The aim is to computethe local Galois cohomology of some two dimensional representations corresponding to thetempered ones via the local Langlands correspondence. In Section 4, we collect some facts
concerning Chow motives, their cohomology theories, and the morphism between the cohomology theories. We are interested in the rational motivic cohomology and the absolute4

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI`-adic cohomology. We refer to Jannsen’s book [17] for cohomology theories and to Scholl’s
article [31] for generalities on Chow motives. In Section 5, we give a proof of Theorem 1.1
following the outline given in this introduction. Sections 5.1, 5.2, 5.3 are devoted to theproof of Propositions 1.4, 1.3 and Theorem 1.1 respectively. Section 7 is independent of
the earlier sections aside from some general notations. The aim is to construct an explicit
family of genus 2 curves satisfying Condition (GL2 ) such that the dimension of the rational
G1 group is arbitrarily large.
Acknowledgment The problem, or the statement of Theorem 1.1 for Drinfeld modular
curves, and the idea of proof to use some of the logic in the paper by Ramakrishnan [25]were suggested by Seidai Yasuda. He also helped us at various other places. We also thankTetsushi Ito for helpful suggestions.
The first author is partly supported by JSPS Grant-in-Aid for Young Scientists (B)No.23740015 and Hakubi project of Kyoto University. The second author was partially
supported by JSPS Grant-in-Aid for Scientific Research 21654002 and by World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan. The third author
is partially supported by JSPS Grant-in-Aid for Scientific Research No.23740027.
2. The construction of abelian varieties associated to automorphic
representations of GL2 in the function field case
The aim of this section is to prove Proposition 2.1. The abelian variety Aπ will be
used throughout this article. This proposition is the function field analogue of a theoremof Shimura [29, p.183, Thms 7.14, 7.15]. The proof is different in that we use the Tate
conjecture (a theorem of Zahrin and of Mori) in this paper. In doing so, we need not use
the theory of modular forms (which does not seem to be available in the literature for thefunction field case) as much as Shimura did. Our construction has a shortcoming that theresulting abelian variety Aπ is neither a sub nor a quotient of the Jacobian of a Drinfeld
modular curve in a canonical way. Another output of this section is Corollary 2.5. This will
be used in the proof of Lemma 5.2.
We fix an isomorphism ι : Q` ∼= C. We use the results from the theory of automorphic
forms with values in C as those with values in Q` via this fixed isomorphism.
2.1. We regard a place of k as a closed point of C and vice versa. Let us choose a closed
point ∞ ∈ C0 and set A = H 0 (C \ {∞}, OC ). For a place ℘ of k, we let k℘ denote thecompletion of k at ℘. 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 of integers ofk℘ .
Let I ⊂ A be a non-zero ideal. We let MI∞ denote the functor which classifies Drinfeld
modules of rank 2 with a Drinfeld full level I structure on a k-scheme. When Spec (A/I)
contains two or more closed points, it is representable by an affine curve over Spec k (see [8,p.576, Proposition 5.3] for construction of the moduli scheme over A; we consider its base∞change to k). We will denote the representing scheme by the same symbol. Let M I denote∞∞the smooth compactification of MI∞ . We set M = lim M I where the limit is taken over←−ideals of A such that Spec (A/I) contains two or more closed points.
b = lim A/I. Let A∞ = Ab ⊗A kLet A = Ak denote the ring of adeles of k. We set A←− ∞denote the ring of finite adeles. The group GL2 (A∞ ) acts on M (see [8, p.577]). Let∞b ⊂ GL2 (A∞ ) be a compact open subgroup. We let M ∞K ⊂ GL2 (A)denoteK = K\Mb → GL2 (A/Ib A)],b then wethe quotient. For example ([8, p.578]), if K = KI = Ker[GL2 (A)∞∞recover M I = KI \M for I as in the previous paragraph. If L is a k-algebra, we write∞∞
M K,L = M K ×Spec k Spec L.
2.2. Let Π denote the set of (isomorphism classes of) cuspidal automorphic representationsof GL2 (A). Let St denote the Steinberg representation of GL2 of a nonarchimedean local
field. We have an isomorphism
M
∞
1
(2.1)
lim
Hét
(M K,ksep , Q` ) ∼(⊗v6=∞ πv ) ⊗ ρ(π)=−→∞
K⊆GL2 (A):open compact subgroupπ∈Π,π∞ ∼=St

THE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

5of GL2 (A∞ ) × Gal(k sep /k)-modules, where ρ(π) is the 2-dimensional representation ofGal(k sep /k) over Q` which is determined by π via the global Langlands correspondence
sep
and whose restriction to Gal(k∞/k∞ ) is isomorphic to Sp. (See [6, Corollary 2.3].) Given ab we may take the K-invariant part of both sides above
compact open subgroup K ⊂ GL2 (A),and obtain (see [10, Section 8] for the description even when K is large)
M
∞
1
(2.2)
Hét
(⊗v6=∞ πv )K ⊗ ρ(π).
(M K,ksep , Q` ) ∼=π∈Π,π∞ ∼=St

For any place ℘ of k, the Galois representation ρ(π)|Gk℘ (or the associated Weil-Deligne
1

representation) corresponds to π℘ ⊗ |det|− 2 via the local Langlands correspondence. (See[8, Theorem A]. See alsoe [20, Corollaire VII.5].)
2.3. Let n be a non-zero ideal of A. Set



a
a b
K1,℘ (n) =∈ GL2 (A℘ )
c
c d

 
b
1
≡
d
0


∗
1


mod n

for a place ℘ 6= ∞ andK1 (n) =Y℘6=∞

K1,℘ (n) =


a
c



a
b
b
∈ GL2 (A)
c
d

 
b
1
≡
d
0


∗
1


mod n .

Let π be a cuspidal automorphic representation of GL2 (A) such that π∞ ∼= St. By [4,K(n )p.302, Theorem 1.4], there exists a non-zero ideal nπ ⊂ A such that dimQ` ⊗℘6=∞ π℘ 1,℘ π =1. (This is the prime-to-∞ part of the conductor of π. The conductor is then nπ ∞ sinceπ∞ ∼= St.) We consider the action of the Hecke operator T℘ on a nonzero element inK(n )⊗℘6=∞ π℘ 1,℘ π for ℘ prime to nπ ∞. We denote the eigenvalue by a℘ (π). Let Kπ denotethe field generated by a℘ (π) over Q where ℘ runs over all primes prime to nπ ∞.
For an abelian variety B, we write T` B for the Tate module, set V` B = T` B ⊗Z` Q` andV` B = V` B ⊗Q` Q` . For a Q` -vector space, we let V ∨ = HomQ` (V, Q` ) denote the dual.
Proposition 2.1.
(1) The field Kπ is a number field. Let I(π) denote the set of embeddings Kπ ,→ Q` .
(2) For any τ ∈ I(π), there exists a cuspidal automorphic representation π τ such thatτ ∼(a) π∞= St,(b) nπ = nπτ ,(c) av (π τ ) = τ (av (π)) for all v prime to nπ ∞.
L(3) There exists a k-simple abelian variety Aπ such that (V` Aπ )∨ ∼= τ ∈I(π) ρ(π τ ) andEndk (Aπ ) ⊗Z Q ∼= Kπ .
Note that by (c) and the strong multiplicity one theorem, π τ is uniquely determined.
Note also that by the Tate conjecture, the abelian variety Aπ is uniquely determined up toisogeny.
2.4. We prove Proposition 2.1 inductively. Fix a nonzero ideal n ⊂ A. The inductive
assumption is that(*n)

There exists an abelian variety Aπ0 satisfying the conditions in Proposition 2.1
for those π 0 such that nπ0 divides n and nπ0 6= n.

Note that (*n) for n = (1) holds trivially. Assuming (*n), we prove the assertions for πsuch that nπ = n.
K(n)
Set dπ,n = dimQ` ⊗℘6=∞ π℘ 1,℘ . Let Πnew
1 (n) denote the (finite) set of cuspidal automorphic representations π such that π∞ ∼Stand dπ,n = 1.
=

6

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

Lemma 2.2. Assume that (*n) holds. There exist abelian varieties over k, denoted JK∞,new
1 (n)and JK∞,old, such that
1 (n)

)∨ ∼
(V` JK∞,new
=
1 (n)

M

ρ(π),

π∈Πnew(n)
1)∨ ∼(V` JK∞,old
=
1 (n)

M

ρ(π)dπ,n .

dπ,n >1

Q
is isogenous to an abelian variety of the form π Aπ (possibly with multiMoreover JK∞,old
1 (n)plicity) where nπ |n and nπ 6= n hold for each π appearing in the product.
Proof. The construction is done inductively. Suppose we are given an abelian variety B
over k such that

 

M
M
(V` B)∨ ∼ρ(π) ⊕ 
ρ(π)bπ =dπ,n =1dπ,n >1

with bπ ≥ 0 for each π such that dπ,n > 1. Note that for π such that dπ,n > 1, we havenπ 6= n (using [4, p.306, Corollary]) so that the inductive hypothesis applies, that is, there
exists an abelian variety Aπ for this π. We have
Homk (B, Aπ ) ⊗Z Q`∼= HomQ` [Gk ] (V` B, V` Aπ ) ⊗Q` Q`∼= HomQ` [Gk ] (V` B, V` Aπ )where the first isomorphism is the Tate conjecture (a theorem of Zarhin [37] and of Mori [24]),and the second is [5, Lemma 29.2, (29.5)]. If bπ ≥ 1, then we can construct a nontrivial
map V` B → V` Aπ (namely, the projection to ρ(π) followed by the canonical inclusion).
Hence there exists a nontrivial morphism B → Aπ , which is surjective since Aπ is simple.
0
Let B 0 denote the connected component of the kernel of this morphism. Then B 0 = Bredis an abelian variety. (This fact is well known when the base field is perfect; we refer to[23, p.1, Theorem 1] for the case when the base field is not necessarily perfect.) Note that
HomQ` [Gk ] (ρ(π), ρ(π 0 )) = 0 for π such that dπ,n = 1 and π 0 such that dπ,n > 1 since ρ(π)
and ρ(π 0 ) then have different conductors and are not isomorphic. It follows that

 

M
M
0
ρ(π) ⊕ 
ρ(π)bπ (V` B 0 )∨ ∼=dπ,n =1P

dπ,n >100π bπ , and B is isogenous to B × Aπ .

P

with π bπ >Now we apply this procedure repeatedly.
To prove the lemma, we start from the Jacobian
P∞
JK∞1 (n) of M K1 (n) and proceed until π b0π = 0. Then we have JK∞,new
as B 0 in the last step
1 (n)of the process. Also one has JK∞,old
as the product of all the Aπ ’s that appeared in the
1 (n)process. It is clear that JK∞,new
and JK∞,oldsatisfy the desired properties.
1 (n)
1 (n)



2.5. We now describe the decomposition of JK∞,new
into the k-simple factors (that is, we
1 (n)find an abelian variety over k isogenous to JK∞,new,whichis a product of k-simple abelian
1 (n)varieties). Each simple factor will be the desired Aπ for some π. We define a Hecke algebra
T ⊂ Endk (JK∞1 (n) ) as the subalgebra generated by the Hecke operators T℘ and S℘ for ℘prime to n. We define Tnew ⊂ Endk (JK∞,new) ⊗Z Q as follows. By Lemma 2.2 and the Tate
1 (n)conjecture, we obtain and fix an isogeny ψ : JK∞,new
× JK∞,old→ JK∞1 (n) . If T ∈ Endk (JK∞1 (n) ),
1 (n)
1 (n)
∞,new
we obtain an element T 0 in Endk (JK1 (n) ) as the composite
ψ

T

ψ∨

JK∞,new→ JK∞,new
× JK∞,old−→ JK∞1 (n) −→ JK∞1 (n) −−→ JK∞,new
× JK∞,old→ JK∞,new
,
1 (n)
1 (n)
1 (n)
1 (n)
1 (n)
1 (n)
where the first arrow is the canonical inclusion, ψ ∨ is the dual isogeny, and the last arrow is
the projection. We define Tnew ⊂ Endk (JK∞,new) ⊗Z Q to be the subalgebra generated by the
1 (n)elements of the form (deg ψ)−1 T 0 . Note that T → Tnew is an algebra homomorphism andthat Tnew is a commutative algebra since T is commutative. By abuse of notation, we letT℘ ∈ Tnew (resp. S℘ ∈ Tnew ) denote the element corresponding to T℘ ∈ T (resp. S℘ ∈ T).

THE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

7

From here, we proceed in a manner very similar to the proof of Proposition 4.2 in [33,p.228]. We have
(2.3)
M
Y
∞,new
ρ(π)∨ ) ∼Endk (JK∞,new) ⊗Z Q` ∼Q`== EndQ` [Gk ] (V` JK1 (n) ) ∼= EndQ` [Gk ] (1 (n)π∈Πnew(n)1π∈Πnew(n)1

where the first isomorphism is due to the Tate conjecture and the last isomorphism follows
from the irreducibility of ρ(πi ) and the fact that if i 6= j then ρ(πi ) and ρ(πj ) are notisomorphic.
(n)2.6. Let fπ∞ denote a nonzero element in ⊗℘6=∞ π℘ 1,℘ . We define a map
M
φ:⊗℘6=∞ π℘K1,℘ (n) → HomQ (Tnew , Q` )K

π∈Πnew(n)
1

by φ(fπ∞ )(T℘ ) = a℘ (π) for ℘ prime to n (this map is independent of the choice of the fπ∞ ).
Lemma 2.3. The map φ is an isomorphism.
Proof. We see from (2.3) that dimQ Tnew is smaller than or equal to the dimension of theleft hand side.
P It suffices to prove the injectivity of φ.
Let f = π∈Πnew (n) cπ fπ∞ with cπ ∈ Q` be an element such that φ(f )(T ) = 0 for all1
T ∈ Tnew . It is enough to show f = 0. For each π and each place ℘ such that ℘ - n, denote
the Satake parameters at ℘ for π by α℘ (π) and β℘ (π). For each ℘ with ℘ - n and r ≥ 1, it
is easy to find an element T℘,r ∈ Tnew such thatφ(fπ∞ )(T℘,r ) = α℘ (π)r + β℘ (π)r
for all π ∈ Πnew
1 (n). Then we haveXcπ (α℘ (π)r + β℘ (π)r ) = 0π∈Πnew(n)
1

for any r ≥ 1. This implies that cπ = 0 for all π ∈ Πnew
1 (n) by [16, p.806, THEOREM].



2.7. Using Lemma 2.3, we obtain Tnew = Endk (JKnew) ⊗Z Q. Since Tnew is commutative
1 (n)and Endk (JKnew) ⊗Z Q is a semisimple Q-algebra, we have Tnew ∼= F1 × · · · × Fs where each
1 (n)Fi is a number field by Wedderburn’s theorem, for some s ∈ Z. We set
Ji = JKnew/Ker[Endk (JKnew) → Fi ]
1 (n)
1 (n)
for each i. Then JKnew(n) is isogenous to J1 × · · · × Js .
1 (n)We haveM
V ` Ji ∼(⊗℘6=∞ π℘K1,℘ (n) ) ⊗ ρ(π)=π∈Πi

for some subset Πi ⊂ Πnew
1 (n). Take a π ∈ Πi . We claim that Ji satisfies the conditions (1),(2) and (3) of Proposition 2.1 so that we may call this Aπ .
K (n)We see from (2.3) that an element of Tnew acts as a scalar on each (⊗℘6=∞ π℘ 1 ) ⊗ ρ(π).
Giving this action is equivalent to giving an algebra homomorphism
pr

ϕπ

i
Tnew −−→Fi −−→ Q`

where pri is the projection to the i-th factor and ϕπ is some embedding.
K(n)From Lemma 2.3, we have dimQ` ⊕π∈Πi ⊗℘6=∞ π℘ 1,℘= dimQ Fi , hence if we fix an0element π ∈ Πi , then any other π ∈ Πi corresponds to the algebra homomorphism
pr

ϕ 0

i
Tnew −−→Fi −−π→ Q`

for some embedding ϕπ0 .
Note that Kπ ⊆ Fi . If Kπ 6= Fi , then there exists some ϕπ0 such that ϕπ |Kπ = ϕπ0 |Kπ .
This contradicts that π and π 0 are not isomorphic using the strong multiplicity one theorem,
hence Kπ = Fi . This completes the proof of Proposition 2.1.


8

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

2.8. We will restate the result obtained in this section in the form to be used elsewhere.
We will use Corollary 2.5 in the proof of Lemma 5.2.
We define an equivalence relation on the set Π∞ of cuspidal automorphic representations
such that π∞ ∼= St by π ∼ π 0 if and only if Aπ is isogenous to Aπ0 .
Q∞
dCorollary 2.4. The Jacobian JK∞1 (n) of M K1 (n) is isogenous to π∈Π∞ /∼ Aππ,n .
Q
Proof. We see from Lemma 2.2 that JK∞1 (n) is isogenous to some product of the form π Aπ(possibly with multiplicity). Then the description of V` Aπ in Proposition 2.1(3) and (2.2)
leads to the claim.
In the following corollary, we are taking ∞ to be ℘.
Corollary 2.5. Let π be a cuspidal automorphic representation. Let Stπ denote the set ofplaces v such that π℘ ∼= St. Then, for each ℘ ∈ Stπ , there exists a Drinfeld modular curve℘℘
MK such that Aπ is a simple factor of JacM K .
Proof. Fix a prime ℘ ∈ Stπ . As we had done in Section 2.3, we can find an ideal n of theK 0 (n)coordinate ring H 0 (C \{v}, OC ) such that dimQ` ⊗℘0 6=℘ π℘01,℘≥ 1. Then by Corollary 2.4,℘

Aπ is a simple factor of JacM K .



3. Galois cohomology of two dimensional representations
The aim of this section is to prove Proposition 3.1. The vanishing result will be used in
the proof of Lemma 5.1. The nonzero result will be used in the proof of Lemma 6.2. For
the application, σ below will be a local component of some 2-dimensional direct factor of
the Jacobian of a curve over k.
3.1. Let F be a nonarchimedean local field of characteristic p ≥ 0. Fix a uniformizer $ ∈ F .
Let GF = Gal(F sep /F ) denote the absolute Galois group. Let ` be a prime number prime
to p. Let σ be a two dimensional `-adic representation, that is, a continuous homomorphism
σ : GF → GL(Vσ ) where Vσ is a two dimensional E-vector space ([E : Q` ] < ∞). Let
Vσ,Q` := Vσ ⊗E Q` . Let WF denote the Weil group and i : WF ,→ GF be the canonicalinclusion. We also denote σ the representation of WF on Vσ,Q` obtained by the restriction.
We write W D(σ) for the Weil-Deligne representation attached to σ (see Bushnell-Henniart[3, p.206, Theorem]) We let π = π(σ) denote the irreducible admissible representationof GL2 (F ) which corresponds to σ via the local Langlands correspondence (see BushnellHenniart [3, p.212, Langlands correspondence]). We fixed an isomorphism ι : Q` ' C. Let
St denote the Steinberg representation of GL2 (F ).
1

Proposition 3.1. Let the notations be as above. Suppose that π(σ) ⊗ |det| 2 is tempered.
Then
(
1
1
1 if π(σ) ⊗ |det| 2 ∼= St and n = 0 or π(σ) ⊗ |det| 2 ∼= St and n = 2,
1
dimE H (GF , Vσ (n)) =0 otherwise.
Lemma 3.2. We havedimE H 1 (GF , Vσ (n)) = −ords=0 L(s, Vσ (n)) − ords=0 L(s, Vσ (n)∗ )where ords=0 means the order of zero at s = 0.
Proof. We havedimE H 1 (GF , Vσ (n))= dimE H 0 (GF , Vσ (n)) + dimE H 2 (GF , Vσ (n))= dimE H 0 (GF , Vσ (n)) + dimE H 0 (GF , (Vσ (n))∗ )= −ords=0 L(s, Vσ (n)) − ords=0 L(s, Vσ (n)∗ ).

The first equality uses that the Euler-Poincare characteristic is zero ([30, p.101, II.5.7.
exercise 4]), the second equality uses the local Tate duality (see [28, Theorem 1.4.1]), andthe third equality follows from the definition of the L-factor.
Lemma 3.3. Let χ : F × → C× be a unitary character. Let n ∈ Z. Then we haven−1 χ = 1 and n = 0,
ords=0 L(s + , χ) =0otherwise.
2

THE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

9nProof. The L-factor has a pole at s = 0 if and only if χ($−1 )|$−1 | 2 = 1 where $ isa uniformizer. If this holds then χ must be unramified. Since χ is unitary, n = 0 andχ($−1 ) = 1.

We compute the L-factor in terms of π(σ) as follows. We haveL(s, Vσ (n))

andL(s, Vσ (n)∗ )= L(s, π(Vσ (n)))
11= L(s, π(σ) ⊗ | det | 2 ⊗ | det |n− 2 )1= L(s + n − 21 , π(σ) ⊗ | det | 2 )= L(s, π(Vσ (n)∗ ))= L(s, π(Vσ (n))∨ (1))
31
= L(s, (π(σ) ⊗ | det | 2 )∨ ⊗ | det |−n+ 2 )1= L(s − n + 23 , (π(σ) ⊗ | det | 2 )∨ ).

We use the classification.
Lemma 3.4. The tempered representations of GL2 (F ) are (i) supercuspidals, (ii) unitary twists of the Steinberg representation, and (iii) unitary principal series representation
2
IndGL
B (χ1 , χ2 ) with unitary characters χ1 , χ2 .
Proof. See Jacquet [14]. See also Kudla [19, p.375, Example 2.5].



3.2. Proof of Proposition 3.1. It suffices to consider the three cases.
1
Case (i)[π(σ)⊗|det| 2 is supercuspidal]: Since the L-factor of a supercuspidal representationis 1, we have L(s, Vσ (n)) = L(s, Vσ (n)∗ ) = 1 in this case. Hence the proposition follows
using Lemma 3.2.
1
Case (ii)[π(σ) ⊗ |det| 2 is a unitary twist St ⊗ χ of the Steinberg representation]: We have
= L(s + n − 21 , St ⊗ χ)
= L(s + n − 12 , χ),∗L(s, Vσ (n) ) = L(s − n + 32 , St∨ ⊗ χ−1 )= L(s − n + 32 , St ⊗ χ−1 )= L(s − n + 32 , χ−1 ),L(s, Vσ (n))

where we used that St ∼= St∨ (See Bushnell-Henniart [3, p.68, (9.10.5)]) and formulas in [19,p.377, 3.1]. The claim follows then from Lemmas 3.2 and 3.3.
12Case (iii)[π(σ) ⊗ |det| 2 is a unitary principal series representation IndGL
B (χ1 , χ2 ) withunitary characters χ1 , χ2 .]: The L-factors are2= L(s + n − 12 , IndGL
B (χ1 , χ2 ))1= L(s + n − 2 , χ1 )L(s + n − 21 , χ2 ),
−1
−1
∗
2
L(s, Vσ (n) ) = L(s − n + 32 , IndGL
B (χ1 , χ2 ))−1
3= L(s − n + 2 , χ1 )L(s − n + 23 , χ−12 ).

L(s, Vσ (n))

The claim follows then from Lemmas 3.2 and 3.3.
4. Motivic cohomology and `-adic cohomology for Chow motives
We collect some definitions and properties concerning Chow motives and cohomology
theories from Jannsen [17] and Scholl [31]. For the application, we take the base field L
below to be either k or k℘ . The Chern class or the map r below will be used only with thebase field k℘ .
4.1. We first recall some definitions and properties of twisted Poincaré duality theories ([2],see [17, Section 6]). For a quasi-projective scheme X over a field L and a closed subschemeZ ⊂ X, the (rational) motivic cohomology of X with support on Z and the (rational)
motivic homology of X are defined to bei
HM,Z(X, Q(j)) = K2j−i (X)(j) ,M
0Ha (X, Q(b)) = Ka−2b(X)(−b) .

We refer to [17, p.104, 6.12] for the precise definitions.

10

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHILet ` be a prime different from the characteristic of L. We will use the (absolute) `-adic
cohomology with support and the (absolute) `-adic homology:i∗Hét,Z(X, πXQ` (j)),ét!
Hi (X, Q` (j)) = H −i (X, πX
Q` (−j))where πX : X → Spec L is the structure morphism. (The reader may find convenient
the torsion coefficient case given in [2, p.193, 2.1], and the geometric case with rationalcoefficient given in [17, p.87, 6.8].)
In [17], Jannsen considered the motivic theory and the geometric (as opposed to theabsolute) `-adic theory, and the morphism between the two twisted Poincaré duality theories.
In a similar manner, one can define a morphism from the motivic theory to the absolute`-adic theory above. We writeiir : HM,Z(X, Q(j)) → Hét,Z(X, Q` (j)),
0Métr : Ha (X, Q(b)) → Ha (X, Q` (b))

for the induced maps. We refer to [17, p.126, (8.4.1)] and the remarks preceding [17, p.125,
8.3 a)] for the details (on the geometric case).
4.2. We will use the setting for Chow motives given by Scholl [31, Section 1]. We recall
some of the notations.
Let L be a field. We let VL denote the category of projective smooth L-schemes. We
let ML denote the category of L-motives for rational equivalence ([31, p.165, 1.4]). Take
X, Y ∈ VL . Suppose X is connected and of pure dimension d. We let
Corrr (X, Y ) = CHr+d (X × Y ) ⊗Z Q
denote the correspondences of degree r. For the definition of correspondences for general
X and the composition ◦ of correspondences, we refer to [31, 1.3]. Recall that an object(X, p, m) ∈ ML is a triple with X ∈ VL , an idempotent p = p2 ∈ Corr0 (X, X), and an
integer m. The hom group HomMk ((X, p, m), (Y, q, n)) is by definition q◦Corrdim X (X, Y )◦p.
We extend the definition of the cohomology groups to objects in ML . Let (X, p, m) ∈aa
ML . Note that p ∈ Corr0 (X, X) defines a map p : HM
(X, Q(n)) → HM
(X, Q(n))) and aaamap pét : Hét
(X, Q` (n)) → Hét
(X, Q` (n)). We refer to [32, Section 1] for the construction
when the cohomology theory is the K-cohomology theory. The procedure works for the twocohomology theories considered here. For integers a, b, we seta,baa((X, p, m)) := Im[p : HM
(X, Q(m + b)) → HM
(X, Q(m + b))],
HM
a,b
a
a
Hét ((X, p, m)) := Im[pét : Hét (X, Q` (m + b)) → Hét
(X, Q` (m + b))].

Then the r and r0 defined in the previous section induce a map
a,ba,br : HM((X, p, m)) → Hét((X, p, m)).

5. Proof of Theorem 1.1
We give a proof of Theorem 1.1. We refer to Section 1 for the outline of proof.
5.1. We begin with the proof of Proposition 1.4.
1
By Assumption (GL2 ), we have Hét
(Bksep , Q` ) ∼= ⊕nj=1 ρj where each ρj is an irreducible
two dimensional representation of Gk with coefficients in Q` .
We consider the case when there does not exist a pair (j, ℘) such that ρj |Gk℘ ∼
= Sp.
2,2 1
Lemma 5.1. In this case, we have Hét(h (Bk℘ )) = 0 for any ℘.

Proof. We use the definition of h1 in [31, p.176, Theorem 4.4]. It says in particular that
there exists a curve Z ,→ B such that the projector to h1 factors as
2,2
2,2
2,2 1
Hét
(Bk℘ ) → Hét
(Zk℘ ) → Hét(h (Bk℘ ))where the first map is the pullback. Using the Leray spectral sequences for Bk℘ and Zk℘ ,we see that it suffices to prove thatqq
Im[H p (Gk℘ , Hét
(Bk℘sep , Q` (2))) → H p (Gk℘ , Hét
(Zk℘sep , Q` (2)))],

THE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

11

where the map is that induced by the pullback map, is zero for each pair (p, q) with p+q = 2.
It is easy to see that the target is zero when (p, q) = (2, 0). When (p, q) = (0, 2), using thefact that Z is a curve, it follows from the weight argument that the target is zero. Suppose(p, q) = (1, 1). Let πj denote the cuspidal automorphic representation corresponding to ρj by
the global Langlands correspondence [9, p.566, Theorem A]. By the Petersson-Ramanujan
1conjecture (see [9, p.566, Theorem B]), we have that each πj,℘ ⊗ | det | 2 is tempered. Since
we know that Sp corresponds to St via the local Langlands correspondence, we obtain1
H 1 (Gk℘ , Hét
(Bk℘sep , Q` (2))) = 0 from Proposition 3.1.
Now suppose that there exists a pair (j, ℘) for which ρj |Gk℘ ∼= Sp. Let π be the unitarycuspidal automorphic representation associated to ρj via the global Langlands correspondence. Since we have π℘ ∼= St, we can apply Proposition 2.1 and find an abelian variety Aπ .
We use the argument in the proof of Lemma 2.2. We have(V` B, V` Aπ ) 6= 0.
Homk (B, Aπ ) ⊗Z Q` ∼= Hom
Q` [Gk ]Therefore there is a nontrivial homomorphism B → Aπ which is an isogeny since both B
and Aπ are simple. We may and will assume that B = Aπ .
Lemma 5.2. For any place ℘ of k there exists a finite dimensional Q-vector space Vπ,℘ ⊂2,2 1
HM(h (Aπ )) such that
(
2,2 1
Hét
(h (Aπ,k℘ )) if ℘0 = ℘,r℘0 (Vπ,℘ ) ⊗Q Q` =0
if ℘0 6= ℘.
Proof. Define Stπ to be the set of places ℘ of k such that π℘ ∼
= St.
2,2 1
(1) Case ℘ ∈/ Stπ . We set Vπ,℘ = 0. Then it suffices to show that Hét(h (Aπ,k℘ )) = 0
for ℘ ∈/ Stπ . This can be proved in the same manner as in the proof of Lemma 5.1, henceis omitted.
℘(2) Case ℘ ∈ Stπ . By Corollary 2.5, Aπ is a k-simple factor of JacM K for some compact
open subgroup K ⊂ GL2 (A℘ ). Here A℘ is the finite adeles as defined in Section 2.1 with ℘in place of ∞ there.
By [18, p.1090, Proposition 9.1], there exists a finite dimensional Q-vector space℘

2,2 1(h (M K ))VK℘ ⊂ HM

such that(5.1)

r℘0 (VK℘ ) ⊗Q Q` =

(
℘
2,2 1
Hét
(h (M K,k℘ )) if ℘0 = ℘,0
if ℘0 6= ℘.

Since r℘0 are compatible with the decomposition of a Chow motive, if we set Vπ,℘ to be
2,2 1
image of VK℘ by the projection to HM(h (Aπ )), the desired conditions are satisfied.
Remark 5.3. In [18], they use the étale Chern class map c2,2 (introduced in [11]) insteadof r℘ . However, it is known that r℘ is two times c2,2 (See [11, p.233, Proof of Proposition
2.35]). Therefore, we can apply the result of [18] to our situation.
This completes the proof of Proposition 1.4.



5.2. We prove Proposition 1.3. To prove this, we decompose the Chow motive h(X) into
smaller pieces as follows. Let us use the formalism of Chow motives as explained in [31].
By Poincaré reducibility theorem (see [23, p.1, Theorem 1] for the case where the basefield is imperfect), the Jacobian JacX of X is canonically (up to isogeny) isogenous to a
product of k-simple abelian varieties: JacX → B1 × · · · × Bs . Then we have
∼ h0 (X) ⊕ h1 (X) ⊕ h2 (X)h(X) =∼ h0 (X) ⊕ h1 (JacX) ⊕ h2 (X)=Qs1∼( i=1 Bi ) ⊕ h2 (X)= h0 (X) ⊕ hLs∼= h0 (X) ⊕ i=1 h1 (Bi ) ⊕ h2 (X),The first isomorphism is that in [31, p.172, 3.2]. The second isomorphism follows from[31, p.178, Proposition 4.5]. The third isomorphism follows from [31, p.182, Corollary

12

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

5.10]. For the fourth isomorphism, recall (see [7, p.216, Theorem 3.1]) that the projector
corresponding to the hi part of an abelian variety is characterizedthat theQs by the fact
multiplication-by-n map acts as ni on it. Now observe that h( i=1 Bi ) ∼= ⊗si=1 h(Bi ) ∼=⊕i1 ,...,is hi1 (B1 ) ⊗ · · · ⊗ his (Bs ). Since h0 (Bi ) ∼= h(Spec k) for all i, the fourth isomorphismholds true.
Let us write YL = Y ×Spec k Spec L for a k-scheme Y and a k-algebra L. To prove
Proposition 1.3, it suffices to show the following three statements:
2,2 0
2,2 0
(1) The map ⊕℘ r℘ : HM(h (X)) → ⊕℘ Hét(h (Xk℘ )) induces a surjection when thesource is tensored with Q` .
2,2 1
2,2 1
(2) The map ⊕℘ r℘ : HM(h (Bi )) → ⊕℘ Hét(h (Bi,k℘ )) induces a surjection when thesource is tensored with Q` for each 1 ≤ i ≤ s.
2,2 2
2,2 2
(3) The map ⊕℘ r℘ : HM(h (X)) → ⊕℘ Hét(h (Xk℘ )) induces a surjection when thesource is tensored with Q` .
Note that h0 (Xk℘ ) ∼= h(Spec k℘ ) and h2 (Xk℘ ) ∼= (Spec k℘ , id, −1) = L (the Lefschetz motive). Hence the target group of the map in (1) and in (3) are zero, so the surjectivity
follows trivially. The statement (2) follows from Proposition 1.4.
This finishes the proof of Proposition 1.3.


5.3. Proof of Theorem 1.1. Let∂℘,Q` : K2 (Xk℘ )Q` → G1 (Xκ(℘) )Q`
be the boundary map at ℘ tensored with Q` and2
c2,2,℘ : K2 (Xk℘ )Q` → Hét
(Xk℘ , Q` (2))

the étale Chern class map (defined in [11]) at ℘ tensored with Q` . As we remarked in
Remark 5.3, the difference between r℘ and c2,2,℘ is a non-zero constant multiple.
Suppose(1) ∂℘,Q` (x) = 0 if and only if c2,2,Q` (x) = 0 for x ∈ K2 (Xk℘ )Q` ,(2) ∂℘,Q` is surjective if and only if c2,2,Q` is surjective.
Then it is easy to see that Theorem 1.1 follows from Proposition 1.3. We prove (1) and (2).
By [18, p.1059, Proposition 3.1], (1) holds true. Let V ⊂ K2 (Xk℘ )Q be a finite dimensional
vector space. Then, using (1), we have dimQ` ∂℘,Q` (V ⊗Q Q` ) = dimQ` c2,2,℘ (V ⊗Q Q` ). Since2
dimQ G1 (Xκ(℘) )Q = dimQ` Hét
(Xk℘sep , Q` (2)) by [18, p.1056, Proposition 2.1], (2) holds true.
This completes the proof.
6. Proof of Corollary 1.2 and the target group
We give a precise form of Corollary 1.2 and its proof in this section. We also computethe dimension of the target group of the boundary map in Theorem 1.1.
6.1.
Corollary 6.1. Theorem 1.1 holds in all of the following cases.
(1) X is an elliptic curve.
∞(2) X is a (compactified) Drinfeld modular curve. That is, when X = M K for some ∞and K as in Section 2.1.
∞
(3) X = MB,K
is the moduli of D-elliptic sheaves of rank 2 with level K-structure. Here,let B be an indefinite quaternion algebra over k and D a maximal order of B. Let∞
K be an open compact subgroup of (B ⊗k Ak )× and MB,K
is the moduli scheme of
D-elliptic sheaves with level K-structure as defined in [21].
(4) X is the curve Xs of Section 7 for s ∈ k such that D(s) 6= 0.
Proof. We check that Condition (GL2 ) holds in each case. It holds trivially in Case (1).
It follows from (2.2) in Case (2). In Case (3), it follows from [21, p.297, Theorem 14.9
and p.300, Theorem 14.12]. For Case (4), it follows from the discussion at the beginning ofSection 7.
THE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

131
6.2. Let X be as in Theorem 1.1. Write Hét
(Xksep , Q` ) ∼= ⊕nj=1 ρj where each ρj is a 2
dimensional irreducible representation of Gk . Let SpX denote the number of pairs (j, ℘)such that ρj |Gk℘ ∼
= Sp.
2,2 1
Lemma 6.2. We have dimQ` ⊕℘∈C0 Hét(h (Xk℘ )) = SpX .

Proof. It suffices to prove this for X = B a simple abelian variety. If SpB = 0, this holds
by Lemma 5.1. When SpB 6= 0, as we have seen in the discussion after Lemma 5.1 that
B = Aπ for some π. In this case, the claim follows from Proposition 3.1.
Corollary 6.3. Let X be as in Theorem 1.1. Then
• dimQ K2 (X)Q ≥ SpX ,2• dimQ HM(X, Q(2)) ≥ SpX .
Proof. This follows from Lemma 6.2, using Theorem 1.1 and Proposition 1.3 respectively.
Consider Case (4) in Corollary 6.1. We show (Proposition 7.1) that, given an integer n,there exists an element s ∈ k such that D(s) 6= 0 and SpXs ≥ 2n. Hence we have
Corollary 6.4. Suppose the characteristic of k is greater than 5. For any n ∈ Z, there
exists a genus two curve X such that K2 (X)Q ≥ n.
Proof. This follows from Corollary 6.3 and Proposition 7.1 immediately.

Note that in [18, p.1054, Corollary 1.3], a similar statement where genus two is replacedby genus one is proved.
7. abelian varieties of GL2 -type
We keep the notations in Section 2. In this section we will be concerned with some classesof abelian varieties over k which admits many endomorphisms. Such abelian varieties are
originally considered by Ribet [27] in which the ground fields are number fields and they
are called abelian varieties of GL2 -type. We will define an analogue of such objects in thefunction fields setting and construct examples of 2-dimensional abelian varieties over k of
GL2 -type which has multiplicative reduction at given finite places.
Let B be an abelian variety over k of dimension g. We say that B is of GL2 -type if there
exist a number field F of degree g over Q and an embedding ι : F ,→ Endk (B) ⊗Z Q as
Q-algebras. As in [27] we can show that the embedding ι is an isomorphism if B is k-simple.
The abelian varieties Aπi of Proposition 2.1 (3) are examples of abelian varieties over k of
GL2 -type.
Let B be an abelian variety over k of GL2 -type with the embedding ι : F ,→ Endk (B)⊗Z Q.
Let ` be a prime number which is different from the characteristic of k. Via ι and the naturalembedding Endk (B) ⊗Z Q ,→ EndQ` [Gk ] (V` (B)), the field F acts on V` B and its action is
non-trivial and faithful because F is a field and it contains 1. Since [F : Q] = g andrankQ` V` B = 2g, we can view V` B as a free F ⊗Q Q` -module of rank 2. Then we have the2-dimensional Galois representation attached to B:
Yρ : Gk −→ AutF ⊗Q Q` (V` B) ' GL2 (F ⊗Q Q` ) =GL2 (Fv )v|`

where Fv is the completion of F at a finiteof F dividing `. We denote by ρv
Y place v prv
the composition of ρ and the projection
GL2 (Fv ) −→GL2 (Fv ). It is easy to see thatv|`

L(s, V` B) =Y

L(s, ρv ).

v|`

We next construct examples of 2-dimensional abelian varieties over k of GL2 -type byusing Brumer-Hashimoto’s family.
Let p > 5 be a prime number and q be a power of p. Let k and A be as in Section 1.2.
Let H(X; a, b, c) be the polynomial defined by (10), p.479 of [13]. For any s ∈ k, we define

14

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

the following polynomial by using H(X; a, b, c):
(7.1)
13
3
1
7
H(x; − , s+1, −1) := x6 −(2s+3)x5 +(s2 +4s+ )x4 +(2s2 −2s− )x3 +(s2 +2s+ )x2 −2sx4424with the discriminantD(s) := s6 (−9 + 1160s + 1512s2 + 992s3 + 48s4 )2 6= 0
with respect to x. For s ∈ k so that D(s) 6= 0, gluing the two finite surjective morphism
f1 : Spec k[x, y]/(y 2 − H(x; − 74 , s + 1, −1)) −→ A1k,x , (x, y) 7→ x and f2 : Spec k[x1 , y1 ]/(y12 −x61 H( x11 ; − 74 , s + 1, −1)) −→ A1k,x1 , (x1 , y1 ) 7→ x1 by the isomorphism (x, y) ←→ (x1 , y1 ) =( x1 , xy3 ), we have a finite surjective morphism Xs −→ P1 . The resulting variety Xs is a
projective smooth curve√ of genus 2 over k. Put Js = JacXs . Then by [13] we have anembedding ι : F = Q( 5) ,→ Endk (Js ) ⊗Z Q. Actually the results of [13] stated for which
ground fields is the rational function fields over Q, but it is easy to check that the algebraic
operations performed there work over any Z-algebra. Hence we can apply the results in [13].
nLet ℘1 , . . . , ℘n be the non-zero prime ideals of A. Take a non-zero element√ s ∈ ∩i=1 ℘i .
Then it is easy to see D(s) 6= 0. For such s and each finite place v|` of F = Q( 5) we attach
the 2-dimensional Galois representation ρs,v : Gk −→ GL2 (Fv ) from Js as above. Then thelocal behavior of ρs,v at each ℘i is as follows.
Proposition 7.1. For each i ∈ {1, . . . , n}, ρs,v |Gk℘ ' Sp.
i

Proof. For simplicity put ℘ = ℘i . Let A℘ (resp. k℘ ) be the completion of A (resp. k) at ℘.
Fix a uniformizer $ of A℘ , put κ := A℘ /$ and q$ = ]κ. Let Wk℘ be the Weil group of k℘(see [35] for basic properties). Fix a lift Frob$ ∈ Wk℘ of the geometric Frobenius element ofGκ . Then each element σ ∈ Wk℘ can be written uniquely as σ = τ · Frobn$ , τ ∈ Ik℘ , n ∈ Z.
Put n(σ) = n for such σ ∈ Wk℘ . Since Wk℘ is dense in Gk℘ , the restriction map fromthe category of continuous Gk℘ -representations to the category of Wk℘ -representations is
injective. So it suffices to prove WD(ρs,v |Gk℘ ) ' WD(Sp).
i
Gluing the two finite surjective morphism
7
fe1 : Xs,1 := Spec A℘ [x, y]/(y 2 − H(x; − , s + 1, −1)) −→ A1A℘ ,x , (x, y) 7→ x
4
and
1
7
fe2 : Xs,2 := Spec A℘ [x1 , y1 ]/(y12 − x61 H( ; − , s + 1, −1)) −→ A1A℘ ,x1 , (x1 , y1 ) 7→ x1
x1
4
by the isomorphism (x, y) ←→ (x1 , y1 ) = ( x1 , xy3 ) , we have a finite surjective morphism
Xs −→ P1A℘ . The resulting variety Xs is a projective flat scheme over A℘ with relative
dimension one. The projectivity follows from the ampleness of the invertible sheaf O(1) on
P1A℘ and III, Excercise 5.8-(d) of [12]. Since the special fiber of Xs is obtained by glueing
the following affine models:1
1
Xs,1 ⊗A℘ κ = Spec κ[x, y]/(y 2 −x2 (x−1)2 (x− )2 ), Xs,2 ⊗A℘ κ = Spec κ[x1 , y1 ]/(y12 −(1−x1 )2 (1− x1 )2 ),2
2
the possible singularities are at the points Pi ∈ Xs,1 corresponding to the ideals (x −e s thei, y, $) for i ∈ {0, 1, 21 }. It is easy to see that Xs is regular at P 21 . We denote by X
e s is a proper strictly
blowing up of Xs along to the ideals (x, y, $) and (x − 1, y, $). Then X
semistable model of Xs over A℘ and its special fiber is obtained by glueing two copies of P1κ
corresponding to two irreducible components of Xs ⊗ κ at three points P0 , P1 , and P 21 andthen replacing Pi by a chain of i + 1 projective lines for i = 0, 1. Hence Xs ⊗ κ is the unionof five divisors D1 , . . . , D5 which are isomorphic to P1κ over κ. Here D1 and D2 correspondto irreducible components of Xs ⊗ κ and D3 (resp. Di , i = 4, 5) is one chain (resp. twochains) between D1 and D2 . These components intersect at κ-rational points.
Henceforth H ∗ means the etale cohomology. To compute the action of Weil group Wk℘of k℘ on V` Js we use the isomorphism Vl (Js ) ' H 1 (Xs,ksep , Q` ) and the weight spectral`5sequence of Rapoport and Zink for the semistable curve Xs . Put D(0) = i=1 Di andTHE RATIONAL K2 OF A GL2 TYPE CURVE IN POSITIVE CHARACTERISTIC

15`
D(1) = i<j Di ∩ Dj . Note that D(1) consists of six κ-rational points. Then by [26] we havethe following spectral sequence:M(−r+2i)H w−r−2i (Dκ, Q` (−i)) =⇒ H w (Xs,k℘sep , Q` ).
E1−r,w+r =i≥max{0,−r}This gives the weight filtration FilWon H w (Xs,k℘sep , Q` ). The essential E1 -terms are asifollows:
E1−1,2 E10,2
E10,1
E10,0

E10,1 .

We can compute the E1 -terms as follows:
E1−1,2 = H 0 (Dκ , Q` (−1)) ' Q` (−1)⊕6 , E10,2 = H 2 (Dκ , Q` ) ' Q` (−1)⊕5 ,
(1)

(0)

(0)

(0)

(1)

1,0
⊕6
0
E10,1 = H 1 (Dκ , Q` ) = 0, E10,0 = H 0 (Dκ , Q` ) ' Q⊕5` , E1 = H (Dκ , Q` ) ' Q` .

For each σ ∈ Wk℘ , since
X
X
n(σ)
n(σ)
n(σ)
(−1)w tr(σ|H w (Xs,k℘sep , Q` )) =(−1)w tr(σ|E1−r,w+r ) = 5−6+5q$−6q$= −1−q$
,
w

r,w
n(σ)

tr(σ|H 0 (Xs,k℘sep , Q` )) = 1, and tr(σ|H 2 (Xs,k℘sep , Q` )) = q$

, we have thatn(σ)
tr(σ|H 1 (Xs,k℘sep , Q` )) = 2 + 2q$.

Since H 1 (Xs,k℘sep , Q` ) is a free F` := F ⊗Q Q` -module of rank 2 and det(σ|H 1 (Xs,k℘sep , Q` )) =
n(σ)

2n(σ)

q$ , the eigenvalues of σ on F` -module H 1 (Xs,k℘sep , Q` ) are 1, q$
modules we have that
H 1 (Xs,k℘sep , Q` )ss ' F` ⊕ F` (−1).

. Therefore as F` [Wk℘ ]-We next compute the monodromy operator N on H 1 (Xs,k℘sep , Q` ) which is defined by
Grothendieck’s monodromy theorem. By weight monodromy theorem for curves, we have
that
W
1
1
sep
sep
(1) N (FilWi H (Xs,k℘ , Q` )) ⊂ Fili−2 H (Xs,k℘ , Q` ),
∼

⊕2
1
⊕2
1
sep
sep
(2) N : grW−→ grW
2 H (Xs,k℘ , Q` ) ' Q` (−1)
0 H (Xs,k℘ , Q` ) ' Q` .
ι

Recall that F acts on H 1 (Xs,k℘sep , Q` )) via the embeddings F ,→ Endk (Js ) ⊗Z Q ,→EndQ` [Gk ] (V` Js ) and V` Js ' H 1 (Xs,k℘sep , Q` )) as Q` [Gk ]-modules. Therefore the action of√
F = Q( 5) on H 1 (Xs,k℘sep , Q` )) commutes with the Galois action, hence F acts also on
W
W
1
1
1
sep
sep
sep
grW
0 H (Xs,k℘ , Q` ) = Fil0 H (Xs,k℘ , Q` ) = Ker(N ) and gr2 H (Xs,k℘ , Q` ). Note thatW
1
gri H (Xs,k℘sep , Q` ) is a free F` -module of rank one for i = 0, 2. So take a basis e2 of
W
1
1
sep
sep
grW
0 H (Xs,k℘ , Q` ) = Fil0 H (Xs,k℘ , Q` ) over F` . Then by (1) as above, N (e2 ) = 0 sinceW1
sep
Fil−2 H (Xs,k℘ , Q` ) = 0. Take a lift e1 ∈ H 1 (Xs,k℘sep , Q` ) of the element corresponding to
e2 via the isomorphism of (2). Then we have N (e1 ) = e2 . Clearly {e1 , e2 } forms a basisof H 1 (Xs,k℘sep , Q` ) over F` . Therefore the matrix representation of N with respect to this
0 1basis is. Hence we have W D(ρv |Gk℘ ) ' W D(Sp).

0 0Remark 7.2. If Js is not k-simple, then Js is isogenous to the product of two elliptic
curves over k which have split multiplicative reduction at each ℘i . In general, it seems to
be difficult to check the k-simpleness of Js for any k. However for a given k we can checkthe k-simpleness of Js as follows. Assume that Xs has good reduction at some non-zero℘ ∈ Spec A. Take a Néron model Js of Js,k℘ := Js ×k k℘ over A℘ (cf. p.12, Definition
1 and p.16, Corollary 2 of [1]). Put Js,κ := Js ×A℘ κ where κ = A℘ /℘. Then we have

16

MASATAKA CHIDA, SATOSHI KONDO AND TAKUYA YAMAUCHI

that EndA℘ (Js ) = Endk℘ (Js,k℘ ). By taking the special fiber, we have a specialization map
sp
Endk℘ (Js ) ⊗Z Q −→ Endκ (Js,κ ) ⊗Z Q. We have the following commutative diagram:Endk (Js ) ⊗Z Q

α/ Endk℘ (Js,k℘ ) ⊗Z Q
EndQ` [Gκ ] (V` Js )sp∼

/ Endκ (Js,κ ) ⊗Z Q .
/ EndQ` [Gκ ] (V` Js,κ )The vertical allows are natural Q-algebra homomorphisms which are determined by functoriality and these are both injective. The homomorphism α is a natural embedding. The bottom arrow is an isomorphism since V` Js ' V` Js,κ as Q` [Gκ ]-modules by proper smooth basechange. So sp is injective. Hence the composition of α and sp is injective. If L(s, V` Js,κ )−1
is an irreducible polynomial in Q[q −s ], then Js,κ is κ-simple by [34], hence Endκ (Js,κ ) ⊗Z Q
is a division field. Hence Js is also k-simple because of the injectivity of sp ◦ α as above.

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Masataka Chida
Department of Mathematics,
Kyoto University,
Kitashirakawa Oiwake-cho,
Sakyo-ku, Kyoto 606-8502 Japan.
The Hakubi Center for Advanced Research,
Kyoto University,
Yoshida-Ushinomiya-cho, Sakyo-ku,
Kyoto, 606-8302, Japan.
email: chida@math.kyoto-u.ac.jp
Satoshi Kondo
Institute for the Physics and Mathematics of the Universe,
University of Tokyo,
5-1-5 Kashiwanoha,
Kashiwa, 277-8583,
Japan
email: satoshi.kondo@ipmu.jp
Takuya Yamauchi
Department of mathematics,
Faculty of Education,
Kagoshima University,
Korimoto 1-20-6, Kagoshima
890-0065, Japan.
e-mail: yamauchi@edu.kagoshima-u.ac.jp