LOGARITHMIC TRANSFORMATIONS OF RIGID ANALYTIC
ELLIPTIC SURFACESKENTARO MITSUI

Contents
1. Introduction
2. Preliminaries
3. Twistings and Logarithmic Transformations
4. Regular One-Forms and Regular Vector Fields
5. Criterion for Algebraicity
6. Algebraic Elliptic Surfaces with κ = 0 and χ = 0
7. Criterion for Non-Algebraicity
References

1
2
3
6
8
9
9
11

1. Introduction
The main purpose of the present paper is to give examples of algebraic
elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by meansof twistings and logarithmic transformations.
First, we construct twistings and logarithmic transformations of families of Tate curves on a disk. These operations come as surgeries of rigid
analytic elliptic fibrations. Although Ueno gave a rough sketch of such
surgeries in characteristic zero in [12, 6], we give precise definitions andproofs in arbitrary characteristics and unify these two kinds of surgery toalgebraize the resulting surfaces.
In particular, we treat logarithmic transformations which produce multiple fibers whose multiplicities are divided by the characteristic of the basefield. When such fibers appear, we obtain plenty of regular one-forms on
the resulting surface.
In the algebraic case, Mumford discovered the following pathology in[10, II]: In the case of positive characteristic, there exists a proper smooth
algebraic surface X such that the dimension h0 (Ω1X ) of regular one-forms isequal to one while the dimension h1 (OX ) of the first cohomology group of
the structure sheaf is equal to zero.
This work was supported by Grant-in-Aid for JSPS Fellows (21-1111).
1

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KENTARO MITSUI

Algebraizing the above rigid analytic elliptic surfaces, we obtain algebraic surfaces X such that the dimension h0 (Ω1X ) is arbitrarily greater thanthe dimension h1 (OX ).
Second, we treat algebraic elliptic surfaces of Kodaira dimension zero
without singular fibers. Bombieri and Mumford classified the possible combinations of multiple fibers of such surfaces in [1]. However, it is unknown
that all of possible combinations actually occur. We construct some casesof known and unknown elliptic surfaces of this type in a unified method.
Finally, we construct non-algebraic rigid analytic elliptic surfaces. In thecase of characteristic zero, the equality h0 (Ω1X ) = h1 (OX ) gives a necessary
and sufficient condition for algebraicity of certain elliptic surfaces. In thecase of positive characteristic, the condition of multiple fibers of algebraic
elliptic surfaces (Theorem 3.3 in [5]) gives a criterion for non-algebraicity.
Even in the situation where the criterion does not work, we give another criterion for non-algebraicity by Grothendieck’s lifting criterion for projective
smooth schemes (Théorème 7.3 in [3, III]).
Notations and Conventions. We fix an algebraically closed complete nonArchimedean valuation field K with a non-trivial valuation. We assume thatrigid spaces and schemes are defined over K. Let us denote the characteristic of K by p ≥ 0. We mainly use the terminologies and notations of [2],[9], [8], and [7].
Acknowledgment. The author thanks Fumiharu Kato for support.
2. Preliminaries
In this section, we review some facts on rigid analytic elliptic fibrations.
We refer to the previous paper [7] in detail.
An elliptic fibration (over a smooth curve) is a triple (X, C, π) where X
is a smooth surface, C is a smooth curve, and π is a proper flat morphismfrom X to C with connected fibers satisfying the following condition. Thereexists a nowhere dense analytic subset C0 of C such that for any p ∈ C −C0 ,the fiber π−1 (p) is a proper smooth curve of arithmetic genus one over K(p).
We define algebraic elliptic fibrations over algebraic smooth curves inthe same way. An (algebraic) proper smooth surface X is called an elliptic
surface if X admits a structure of (algebraic) elliptic fibration. The analytification of an algebraic elliptic fibration over an affinoid algebra A is anelliptic fibration over Sp A.
Let us denote the genus of a proper smooth curve C by g(C). For an
elliptic fibration (X, C, π), let us denote the length of the torsion submoduleof R1 π∗ OX by l(π). We have the canonical bundle formula and Noether’s
formula.

LOGARITHMIC TRANSFORMATIONS OF RIGID ANALYTIC ELLIPTIC SURFACES

3

Proposition 2.1. Let π : X → C be an elliptic surface. Then there exists a
canonical isomorphism
KX  π∗ L ⊗OX OX (D)where the line bundle L on C and the divisor D on X satisfy the following
conditions:(1) The degree of L is equal to χ(OX ) + l(π) + 2g(C) − 2.
(2) Let C0 be the subset of points on C where the fiber is not smooth.
We write the multiplicity mq and the fiber mq Dq at q ∈ C0 . Then thedivisor D is given by the following equality:XD=
c q Dq .
q∈C0Here, each integer cq satisfies the inequalities 0 ≤ cq < mq .
Remark. If the fiber is tame at q ∈ C0 , then the integer cq is equal to mq − 1.
If all the fibers are tame, then the equality l(π) = 0 holds. This is the casewhen the characteristic p is equal to zero.
Proposition 2.2. Let π : X → C be an elliptic surface. Then the Euler
characteristic χ(OX ) of the structure sheaf is equal to zero if and only if the
fibration π is free from singular fibers.
3. Twistings and Logarithmic Transformations
In this section, we construct twistings and logarithmic transformations.
We treat these two kinds of surgery in the same method. Since we use Tate’s
uniformization of elliptic curves, we only treat families of elliptic curves
with j-invariants whose absolute values are greater than one. Note that,comparing to the logarithmic transformations of complex analytic elliptic
surfaces ([6, 9]), we treat the restricted cases in the rigid analytic case.
Put C := Sp KhT i. Assume that the elliptic fibration π : X → C admits
a relative uniformization in the following sense: There exists an analytic
function α(T ) on C satisfying the following conditions. Put X := A1×
K × C.
(1) The inequalities 0 < α(t) < 1 hold for all t.
(2) The quotient of X by the automorphism (x, t) 7→ (α(t)x, t) is isomorphic to X over C. Note that we have the fundamental domain{(x, t) ∈ X | α(t) ≤ |x| ≤ 1} for this action.
In this case, we obtain the commutative diagram:
φ

/X

??
??  
π
ψ ??


X?

C.

4

KENTARO MITSUI

We take integers a and m such that the conditions gcd(a, m) = 1, a , 0,and m > 0 are satisfied. Choose integers b and n such that the equalityab − mn = 1 holds. For a positive real number ǫ, put
1b mYǫ := {(y, u) ∈ A1×
K × AK | |y u | ≤ ǫ}.

Further, put
Y := Y1 ,∗Y := {(y, u) ∈ Y | u , 0},
C ∗ := C − {0},andX∗ := ψ−1 (C ∗ ).
We may define the isomorphism X∗ → Y∗ by the equalities:
x = yn ua ,t = yb u m .
Let Y be the quotient of Y by the automorphism:(y, u) 7→ α(yb um )−m y, α(yb um )b u .
Note that we have the fundamental domain{(y, u) ∈ Y | |α(yb um )|m ≤ |y| ≤ 1}
for this action. We define the fibration τ : Y → C by a projection (y, u) 7→
yb um . Put Y ∗ := τ−1 (C ∗ ). Let us denote the restrictions of π and τ to thepreimages of C ∗ by π∗ : X ∗ → C ∗ and τ∗ : Y ∗ → C ∗ respectively.
We construct the isomorphism µ : Y ∗ → X ∗ over C ∗ in the following way.
We define relative Tate’s elliptic functions on Y by the equalities (cf., [11]):∞∞X
X
α(t)i xti−
2
Xα (x, t) :=(1 − α(t)i x)2(1 − ti )2
i=−∞
i=1
and
Yα (x, t) :=

∞
X

∞
X
(α(t)i x)2
ti
+
.
i x)3
i )2
(1
−
α(t)
(1
−
t
i=−∞
i=1

These analytic functions give the morphism X∗ → PC2 ∗ by the correspondence:(x, t) 7→ ((Xα (x, t) : Yα (x, t) : 1), t) .
∗Let Z be the image of this morphism. Then the morphism induces an isomorphism X ∗ → Z ∗ . We may define a morphism Y∗ → Z ∗ by the correspondence:(y, u) 7→ (Xα (yn ua , yb um ) : Yα (yn ua , yb um ) : 1), yb um .

LOGARITHMIC TRANSFORMATIONS OF RIGID ANALYTIC ELLIPTIC SURFACES

5

This morphism induces an isomorphism Y ∗ → Z ∗ . Compositing the lastisomorphism Y ∗ → Z ∗ and the inverse of the first isomorphism X ∗ → Z ∗ ,we obtain the isomorphism µ : Y ∗ → X ∗ over C ∗ .
As a result, the fibration τ : Y → C satisfies the following conditions.
(1) The fibration τ is an elliptic fibration with only one tame multiple
fiber over the origin. The multiplicity of the multiple fiber is equal
to m.
(2) The fibration Y ∗ is isomorphic to the fibration X ∗ over C ∗ . The isomorphism is induced by µ.
The construction of the fibration τ dose not depend on the choice of b andn.
For the relative uniformization ψ : X → C of the elliptic fibration
π : X → C, let us denote the fibration τ : Y → C by L(0, m, a) X → C.
When the positive integer m is equal to one, we call the surgery the twistingof the elliptic fibration π. Otherwise, we call the surgery the logarithmictransformation of the elliptic fibration π.
Assume that the elliptic fibration π admits another relative uniformization:
X′ @

φ′

/X
@@


@@  
@
π
ψ′ @@


C.
We take Y and construct Y in the same way. Then the parameter (x′ , t′ ) of
X′ and the parameter (y′ , u′ ) of Y′ satisfy the following equalities:′′

x′ = y′n u′a ,t′ = y′b u′m .
Suppose that the morphism λ : X → X′ , (x, t) 7→ (β(t)x, ρ(t)) is an isomorphism where β(t) is a nowhere-zero analytic function on C and ρ is an
automorphism of C. Then the diagram
λ

/ X′
??
~
~~
?? 
~~ ′
~
φ ??
~~~ φ

X?

X

is commutative. For a sufficiently small positive real number ǫ, we may
define the nowhere-zero analytic function γ := ρ(t)/t for all t such that|t| ≤ ǫ. We define the isomorphism e
λ : Yǫ → Yǫ′ over C by the equality:eλ(y, u) := β(yb um )−m γ(yb um )a y, β(yb um )b γ(yb um )−n u .
Then the isomorphism e
λ induces the isomorphism Y → Y ′ over C.

6

KENTARO MITSUI

4. Regular One-Forms and Regular Vector Fields
In this section, we calculate the dimensions of cohomology groups of theregular one-forms and regular vector fields on the modified elliptic surfaces.
Let α is an element of K such that the inequalities 0 < |α| < 1 hold. Let
1×
C be a smooth curve. Put E := A1×
K /hαi, X := AK × C, and X := E × C.
We consider the surgery in the previous section and use the notations in theprevious section.
First, we assume that the curve C is isomorphic to the disk Sp KhT i.
Since we have the isomorphismdx⊕ π∗ ΩC1 ,Ω1X  π∗ OCx
we obtain the isomorphism:dx
φ∗ Ω1X  ψ∗ OC⊕ ψ∗ ΩC1 .
x
Thus, we may write a global section of this sheaf on X∗ in the following
way:
dx
F(t)
+ G(t) dt.
x
The global section may be written as
!
dy
du
b m
+ G(yb um )(byb−1 um dy + myb um−1 du)+aF(y u ) nyu
on Y∗ . We assume that the global section extends to the whole space Y.
If the characteristic p divides m, then the function F(t) has an at least firstorder zero, and the function G(t) has an at most first order pole at the origin.
If the characteristic p does not divide n, then the function F(t) is analytic
at the origin. In this case, if the value F(0) is zero, then the function G(t)is analytic at the origin. Otherwise, the function G(t) has a first order polewith the residue −aF(0)/m at the origin.
In the same method, we may describe a global section of Ω1X ⊗ OX ((m −1)D0 ) on Y. We write a global section of this sheaf on X∗ in the following
way:
dx
F(t)
+ G(t) dt.
x
We assume that the global section extends to the whole space Y. Then thefunction F(t) is analytic at the origin, and the function G(t) has an at mostfirst order pole at the origin.
Next, we assume that C is a proper smooth curve of genus g(C). Let S bea finite subset of the points of C. For each element q of S , we take integers
aq and mq such that the conditions gcd(aq , mq ) = 1, aq , 0, and mq > 0
holds. Put T := {q | p divides mq }. If the characteristic p is equal to zero,LOGARITHMIC TRANSFORMATIONS OF RIGID ANALYTIC ELLIPTIC SURFACES

7

then the set T is empty. For all elements q of S , we apply L(q, mq , aq ) to
X. Let us denote the resulting elliptic fibration by X({(q, aq /mq )}q∈S ). Put
Y := X({(q, aq /mq )}q∈S ).
Theorem 4.1. If the set T is non-empty, then the equality h0 (Ω1Y ) = g(C) +♯T − 1 holds.
Proof. The above calculations show that the equalityh0 (Ω1Y ) = h0 ΩC1 ⊗ OC (D)P
holds where the divisor D is given by q∈T q. This proves the theorem.

Assume that the set T is empty. We consider the first Cousin problem of
the rational differential one-forms on the proper smooth curve C. The sheaf
exact sequence
0 −→ KC −→ MC ⊗ KC −→ (MC /OC ) ⊗ KC −→ 0.

gives the connecting homomorphism δ : H 0 (C, (MC /OC )⊗KC ) → H 1 (C, KC ).
We put
X aq
θ :=
dtq
mq tq
q∈S
in H 0 (C, (MC /OC ) ⊗ KC ) where tq is a parameter at the point q. Put
X aq
η :=
mq
q∈S
in K. Then the image of δθ under the residue map H 1 (C, KC ) → K is equal
to η.
Theorem 4.2 (cf., Lemma 6.1 in [12]). If the set T is empty, then the equality
η , 0,
g(C),01
h (ΩY ) = 
g(C) + 1, η = 0
holds.

Theorem 4.3. If the base space C is isomorphic to the projective line, thenthe equality♯S ≤ 3,0,h2 (ΘY ) = ♯S − 3, otherwise
holds.

8

KENTARO MITSUIProof. Since the base space is isomorphic to the projective line, there exists
a canonical isomorphism


X

KY  τ∗ OP1K (−2) ⊗ OY  (mq − 1)Dq q∈S

where the divisor mq Dq is the multiple fiber over q. Calculation of h0 (Ω1Y ⊗KY ) and the Serre duality theorem show the theorem.
Remark. In this case, in the case of positive characteristic, if the elliptic
surface Y is algebraic and if the inequality ♯S ≤ 3 holds, then the surface isliftable to characteristic zero.
5. Criterion for Algebraicity
We study the elliptic surface Y over the proper smooth curve that is constructed in the previous section. We use the notations in the previous section.
Let l be the least common multiple of the integers {mq }q∈S . We define thedivisor D on C by the equality:
X aq l
· q.
D :=
mq
q∈S
Theorem 5.1. If there exists an integer m such that the divisor mD is principal, then the elliptic surface Y is algebraic.
Proof. The above condition shows that there exists a rational function f on
C such that the divisor ( f ) is equal to mD. Put M := lm. We define thedivisor D∗ on the relative uniformization A1×
K × (C − S ) by the equalityM∗∗x − f . Put D := φ(D ). Then the image D∗ is a divisor on the preimageof C − S under τ. We show that the divisor D∗ extends to the whole rigid
analytic space Y. For q ∈ S , we take parameters (x, t) and (y, u) for thesurgery L(q, mq , aq ). Take integers b and n such that aq bq − mq nq = 1. We
may write f (t) = taq M/mq g(t) on a neighborhood of q where g(t) is a nowherezero analytic function. Then the divisor D∗ extends to the divisor D on Y
by the equality:
x M − f (t) = ynq M uaq M 1 − y M/mq g(ybq umq ) .
Thus, the divisor D∗ extends to the divisor D on Y. Since the image τ(D) isequal to C, Lemma 4.4 in [7] proves the theorem.
Remark. In this case, if the set T is empty, then the equality h0 (Ω1Y ) =g(C) + 1 holds.

LOGARITHMIC TRANSFORMATIONS OF RIGID ANALYTIC ELLIPTIC SURFACES

9

6. Algebraic Elliptic Surfaces with κ = 0 and χ = 0
We construct algebraic elliptic surfaces τ : Y → C of Kodaira dimension
zero without singular fibers by Theorem 5.1. The notation (c1 /m1 , . . . , cn /mn )describes the multiple fibers of the fibration τ. The integer mi is the multiplicity of the i-th multiple fiber. The integer ci is the coefficient of thedivisor in the canonical bundle formula. The symbol ∗ over mi means thatthe i-th multiple fiber is wildly ramified.
From now on, we assume that the curve C is isomorphic to the projective
line. By Theorem 5.1, we obtain some examples.
Example 6.1. We obtain the algebraic elliptic surface
X((0, −1/2), (1, 1/3), (∞, 1/6))
with κ = 0 and χ = 0. The combination of the multiple fibers is of type(1/2, 2/3, 5/6). In the same way, we obtain the algebraic elliptic surfaces
with the desired invariants whose multiple fibers are of type (1/2, 3/4, 3/4),(2/3, 2/3, 2/3), and (1/2, 1/2, 1/2, 1/2).
Example 6.2. Let λ be a point on C which is not equal to −1, 0, 1, or ∞.
We obtain the algebraic elliptic surface
X((0, 1/2), (1, 1/2), (λ, 1/2), (λ + 1, 1/2), (∞, −2/1))
with κ = 0 and χ = 0. The combination of the multiple fibers is of type(1/2, 1/2, 1/2, 1/2). Assume that the characteristic p is equal to two. By thequotient as in Example 8.1 in [5], we obtain the algebraic elliptic surface
with the same invariants. The combination of the multiple fibers is of type(1/2∗ , 1/2, 1/2).
Example 6.3. Let λ be a point on C which is not equal to 0, 1, or ∞. We
obtain the algebraic elliptic surface
X((0, 1/3), (1, 1/3), (λ, 1/3), (∞, −1/1))
with κ = 0 and χ = 0. The combination of the multiple fibers is of type(2/3, 2/3, 2/3). Assume that the characteristic p is equal to three. Put λ :=2. By the quotient as in Example 8.1 in [5], we obtain the algebraic elliptic
surface with the same invariants. The combination of the multiple fibers isof type (1/3∗ , 2/3).
7. Criterion for Non-Algebraicity
In this section, we give criterions for non-algebraicity of proper smoothrigid analytic surfaces and examples of non-algebraic elliptic surfaces.

10

KENTARO MITSUIg(C) l(τ) multiple fibers00 (1/2, 2/3, 5/6)

0

1

p
2
3
, 2, 3(1/2, 3/4, 3/4)
2
,2
(2/3, 2/3, 2/3)
3
,3
(1/2, 1/2, 1/2, 1/2)
2
,2
(0/2∗ , 1/2, 1/2)
2∗
(1/2 , 1/2)
2
(1/3∗ , 2/3)
3∗
(1/4 , 3/4)
2
(2/4∗ , 1/2)
2∗
(2/6 , 2/3)
2
(3/6∗ , 1/2)
3
a∗(0/p ),0

examples[1, 3][1, 3][4, 5.2][1, 3][1, 3]

unknown[4, 5.2]
unknown
unknown[5, 8.4][5, 8.4]
02[1, 3], [5, 8.1] (somecases)a∗b∗(0/p , 0/p ),0[1, 3] (some cases)
1
0 noneproduct of two elliptic curves
Table 1. The possible invariants of elliptic surfaces τ : Y →C of Kodaira dimension zero without singular fibers. The
symbol means that we treat the type in the present paper.

Proposition 7.1 (cf., Theorem 27 in [6]). Assume that the characteristic pof the base field K is equal to zero. Suppose that the element η in Section 4
is not equal to zero. Then the surface Y in Section 4 is not algebraic.
Remark. If the genus of the base curve is equal to zero, then Theorem 5.1
and the above proposition give the necessary and sufficient condition for the
algebraicity of the surface Y (cf., Appendix 1 in [5]).
Proof. Since the characteristic p is equal to zero, for any proper algebraic
surface Z, the equality h1 (OZ ) = h0 (Ω1Z ) holds. Since we have the equality
h0 (KY ) = g(C), by the Serre duality theorem and by the equality χ(OY ) = 0,
we obtain the equality h1 (OY ) = g(C) + 1. Thus, Theorem 4.2 shows the
proposition.
LOGARITHMIC TRANSFORMATIONS OF RIGID ANALYTIC ELLIPTIC SURFACES 11

Lemma 7.2. Assume that the characteristic of the base field K is positive.
Then any proper smooth algebraic surface Z with the invariants χ(OZ ) = 0,h0 (KZ ) = 0, and h0 (Ω1Z ) = 0 is non-liftable to characteristic zero.
Proof. Assume that the surface Z admitted a lifting. Let Y be the genericfiber of this lifting. The constancy of the Euler characteristic and the uppersemicontinuity of the dimension of the cohomology groups of coherent
sheaves on a proper flat family of schemes shows that the equalities χ(OY ) =0, h0 (KY ) = 0, and h0 (Ω1Y ) = 0 hold. The Serre duality theorem (Theoremin [13, 5.1]) shows that the equality h0 (KK ) = 0 holds. Therefore, we obtain the inequality h0 (Ω1Y ) < h1 (OY ). Since the inequality dose not hold incharacteristic zero, this is a contradiction.

The above lemma, the canonical bundle formula, and Noether’s formula
give the following proposition.
Proposition 7.3. Assume that the characteristic of the base field K is positive. Then any algebraic elliptic surface Z over the projective line without
singular fibers with the invariant h0 (Ω1Z ) = 0 is non-liftable to characteristic
zero.
The above proposition and Grothendieck’s lifting criterion (Théorème
7.3 in [3, III]) show the following proposition.
Proposition 7.4. Assume that the characteristic of the base field K is positive. Then any elliptic surface Z over the projective plane without singularfibers with the invariants h0 (Ω1Z ) = 0 and h2 (ΘZ ) = 0 is not algebraic.
The condition Ui (Theorem 3.3 in [5]) gives a criterion for non-algebraicity
of some elliptic surfaces. Here, we give an example of non-algebraic elliptic
surfaces such that the criterion does not work. Let q and r be two distinct
points on the projective line C. Assume that the integers mq and mr arelarger than one and not divided by p. Put aq = 1 and ar = mr − 1. Then
we can not conclude that the resulting surface Y is not algebraic only by
the conditions U1 and U2 . The above proposition proves, however, that thesurface Y is not algebraic.
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vol. 261, Springer-Verlag, Berlin, 1984, A systematic approach to rigid analytic geometry.

12

KENTARO MITSUI

3. A. Grothendieck, Revêtements étales et groupe fondamental, Springer-Verlag, Berlin,
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88 (1966), 682–721.
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, Criterion for minimality of rigid analytic surfaces, preprint.
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9.
, Minimal models of rigid analytic spaces, preprint.
10. D. Mumford, Pathologies of modular algebraic surfaces, Amer. J. Math. 83 (1961),
339–342.
11. J. Tate, A review of non-Archimedean elliptic functions, Elliptic curves, modular forms,& Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, I, Int. Press, Cambridge, MA, 1995, pp. 162–184.
12. K. Ueno, Compact rigid analytic spaces with special regard to surfaces, Algebraic
geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam,
1987, pp. 765–794.
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no. 2, 219–235.
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502,
Japan
E-mail address: mitsui@math.kyoto-u.ac.jp