Analytic Variation of p-adic Abelian Integrals

In Ann. of Math. 121 (1985), 111^168, Coleman de¢nes p-adic Abelian integrals on curves. Given a family of curves X/S, a differential o and two sections s and t, one can de¢ne a function lo on S by lo…P† ˆ R t…P† s…P† oP. In this paper, we prove that lo is locally analytic on S. Mathematics Subject Classi¢cations (2000). Primary 14G20; Secondary 14D10, 11G20. Key words. p-adic Abelian integrals, algebraic families of curves, locally analytic variation. 1. Introduction Let p be an odd prime number. Let K be a ¢nite extension ofQp, and let R be its ring of integers. Let k be the residue ¢eld of R. Let S be a smooth af¢ne curve over R. Let X ! S be a family of curves over S; in other words, suppose that X=S is proper and smooth of relative dimension one. Let s and t be sections of this family, and set D ˆ s…S† [ t…S†. Given a family of relative differentials of the second kind o on X=S whose polar divisor does not meet the images of s and t, we would like to study the p-adic integrals (as de¢ned by Coleman in [1]) of o from s to t on the ¢bers of the family X=S. One possible de¢nition of these integrals is the following. For each P 2 S, the ¢ber XP above P is a smooth curve, s…P† and t…P† are points of XP and oP is a differential of the second kind on XP, so we can use the construction of [1] to de¢ne lo…P† by lo…P† ˆ R t…P† s…P† oP: We view this as giving a function lo on S. Some interesting arithmetical properties of the family X=S can be phrased in terms of lo. For example, the results of [1] imply THEOREM 1.1. The divisor class of …s…P†† ÿ …t…P†† in XP is torsion if and only if lo…P† ˆ 0 for all o 2 G…S; p OX=S†. However, given the above de¢nition, there is no reason to expect that lo has any good properties at all, since it is a priori only a set-theoretic function. Zarhin gives an alternative construction of p-adic Abelian integrals in [4], but this constuction does not lend itself to studying families. In this paper,we establish that lo is in fact locally analytic on S. By `locally analytic,' we mean that lo is given by a convergent power series on each residue class of S. Compositio Mathematica 124: 57^63, 2000. 57 # 2000 Kluwer Academic Publishers. Printed in the Netherlands. https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1002432618607 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 09 Sep 2017 at 02:15:42, subject to the Cambridge Core terms of use, available at 2. Restriction to Residue Classes To make our results precise, we must formalize the notion of restricting to a residue class of S. The completion of S along a k-valued point P0 is isomorphic to SpecR‰‰T ŠŠ (for some non-canonical choice of a local parameter T ). We view base change by the map SpecR‰‰T ŠŠ ! S as restricting to the residue class of P0. A function on S can be pulled back to a function of T on the residue class. The integrals lo will have denominators of p and hence will not be in the ring R‰‰T ŠŠ, so we introduce KffTgg, the ring of power series which are convergent in the open disk of radius 1.More precisely, KffTgg consists of series Pniˆ0 aiT i such that lim i!1 jaijr ˆ 0 for every real number 0W r < 1. One may regard KffTgg as the ring of rigid analytic functions on an open unit ball. Our main result can now be stated: THEOREM 2.1. Let X=S be a family of curves ando a family of differentials as in the introduction. On any residue class with local parameter T, the integral lo, viewed as a function of T, is an element of KffTgg. The proof of this theorem will be divided into two cases. First, we will give a proof based on crystalline cohomology for the residue classes where the sections do not meet mod p. Then we will give a fairly elementary proof for residue classes where the two sections s and t are congruent mod p. 3. Disjoint Sections To prove the analyticity of integrals on residue classes where the two sections do not meet, we will use the language of crystalline cohomolgy. We will follow the notation for F -crystals used in [3]. We wish to integrate differentials of the second kind on X=S. However, the differential of a function that vanishes on D should integrate to zero. Thus we may view the objects we are integrating as differentials of the second kind modulo differentials of functions that are zero on D, i.e. as classes from H1 DR…X=S;D†. An integral should assign a function on the base to each such class. The problem of integration therefore amounts to ¢nding a section s of the dual of H1 DR…X=S;D†, namely H1 DR……X nD†=S†, such that ho; si ˆ lo. The cohomology modules H1 DR……X nD†=S† and H1 DR…X=S;D† are F -crystals on S (see [2]). Brie£y, this means that they are S-modules with an integrable, convergent connection and an action of Frobenius; see [3] for more details. We will show that the following properties determine a locally analytic section s and also characterize Coleman's integrals: (1) dho; si ˆ hro; si. (2) hdG; si ˆ t Gÿ s G for G a function regular on D. 58 ROLAND DREIER https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1002432618607 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 09 Sep 2017 at 02:15:42, subject to the Cambridge Core terms of use, available at (3) Fs ˆ ps, where F denotes the Frobenius endomorphism. 3.1. CONSTRUCTION OF INTEGRALS The section s will be a locally analytic (Coleman uses the term ``£abby'') section of the cohomology sheaf; in other words, s will be given locally as a section of the pullback of the cohomology to each residue class, with no relation required between residue classes. We have the following: THEOREM 3.1. For each k-valued point P0 of S such that s and t do not meet above P0, there is a unique section s of H1 DR……X nD†=S† restricted to the residue class above P0 satisfying the following conditions: (1) dho; si ˆ hro; si for all o 2 H1 DR…X=S;D†. (2) hdG; si ˆ t Gÿ s G for G a function on X regular on D. (3) Fs ˆ ps, where F denotes the Frobenius endomorphism of H1 DR……X nD†=S†. Proof. Choose a local parameter T for the residue class above P0. We will use the notationÿ KffTgg to denote the pullback of an S-module to the residue class above P0. Let H be H1 DR……X nD†=S† KffTgg. We seek a section s of H. First, the pairing on cohomology is compatible with the connections, which means that for any sections o and s, ho;rsi ‡ hro; si ˆ dho; si: Condition 1 in the statement of the theorem may then be understood as requiring that ho;rsi ˆ 0 for all o, i.e. as requiring that s be horizontal. Therefore, we must ¢nd a vector with the desired properties in the ¢nite-dimensional K-vector space of horizontal sections of H. For the remainder of the proof, we restrict to the residue class above P0. There is a horizontal exact sequence 0! H1 DR…X=S† KffTgg ! H ÿ! Res H0 DR…D†0 KffTgg ! 0; where H0 DR…D†0 denotes the degree 0 part of H0 DR…D† (the residues of a differential form must sum to 0). Using Proposition 3.1.2 of [3], we obtain a corresponding exact sequence of horizontal sections 0! …H1 DR…X=S† KffTgg†r ! Hr ! …H0 DR…D†0 KffTgg†r ! 0: This is an exact sequence of K-vector spaces. However, since all three vector spaces arise as spaces of horizontal sections of F -crystals, they are all equipped with a s-linear endomorphism (where s denotes the Frobenius automorphism of K) which ANALYTIC VARIATION OF p-adic ABELIAN INTEGRALS 59 https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1002432618607 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 09 Sep 2017 at 02:15:42, subject to the Cambridge Core terms of use, available at we will call the action of Frobenius and write as F. A priori this endomorphism depends on a choice of lift of Frobenius, but convergence shows that every lift in fact gives the same endomorphism. Let n be the integer such that sn is the identity on K . Then Fn is a K-linear endomorphism of all the vector spaces in the exact sequence above, and the maps of the exact sequence respect this map. …H0 DR…D†0 KffTgg†r is a one-dimensional K-vector space where Fn acts as multiplication by pn. …H1 DR…X=S† KffTgg†r is a 2g-dimensional K-vector space where Fn acts with eigenvalues of complex absolute value pn=2 (by comparison with crystalline cohomology and the Riemann hypothesis). Since these eigenvalues have different complex absolute value, this extension of vector spaces splits naturally in a unique Fn-invariant way. Condition 2 speci¢es the image of s in …H0 DR…D†0 KffTgg†r, namely that it should have residue ‡1 on t and residue ÿ1 on s. Condition 3 gives the action of F, which means that s must actually be the unique preimage of these residues coming from the Frobenius-invariant splitting (and upon which Fn acts as multiplication by pn). 3.2. COMPARISON WITH COLEMAN'S INTEGRALS We would now like to show that the integrals constructed in the proof of Theorem 3.1 agree with the integrals constructed in [1]. As in the introduction, de¢ne a function lo on S by lo…P† ˆ R t…P† s…P† oP, where the integral is to be interpreted as in [1]. We now prove the following: THEOREM 3.2. Let P be a point of S. If s is the locally analytic section of H1 DR……X nD†=S† constructed in Theorem 3.1 and o is a section of H1 DR…X=S;D†, let sP and oP denote their pullbacks to the ¢ber above P. Then hoP; sPi ˆ lo…P†. Proof. Let P0 be the reduction of P. Since everything is at most locally analytic, we will restrict to the residue class above P0. In particular, for the remainder of this proof, we will write S and X for the restriction of these objects to the residue class of P0. Let n be the positive integer such that P0 is ¢xed by the nth power of Frobenius. There is a commutative diagram of af¢noids The reduction of this diagram commutes with ~ f, the nth power of the Frobenius map 60 ROLAND DREIER https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1002432618607 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 09 Sep 2017 at 02:15:42, subject to the Cambridge Core terms of use, available at