Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers

Bannai, Kenichi; Kobayashi, Shinichi

doi:10.1215/00127094-2010-024

Cited by 22 publications

(55 citation statements)

References 32 publications

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“…The Kronecker theta function is known to satisfy a distribution relation [BK10b]. The aim of this appendix is to prove a similar distribution relation for the underlying Kronecker section.…”

Section: P-adic Interpolation Of P-adic Eisenstein-kronecker Seriesmentioning

confidence: 95%

Eisenstein–kronecker Series via the Poincaré Bundle

Sprang

2019

Forum of Mathematics, Sigma

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A classical construction of Katz gives a purely algebraic construction of Eisenstein-Kronecker series using the Gauß-Manin connection on the universal elliptic curve. This approach gives a systematic way to study algebraic and p-adic properties of real-analytic Eisenstein series. In the first part of this paper we provide an alternative algebraic construction of Eisenstein-Kronecker series via the Poincaré bundle. Building on this, we give in the second part a new conceptional construction of Katz' two-variable p-adic Eisenstein measure through p-adic theta functions of the Poincaré bundle.

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Section: P-adic Interpolation Of P-adic Eisenstein-kronecker Seriesmentioning

confidence: 95%

Eisenstein–kronecker Series via the Poincaré Bundle

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2019

Forum of Mathematics, Sigma

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“…Then by [BK2] Proposition 2.1, the Taylor series of θ z 0 (z) at z = 0 has algebraic coefficients. Note that from the definition, we have…”

Section: P-adic Kronecker Limit Formulasmentioning

confidence: 96%

p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

Bannai¹,

Furusho²,

Kobayashi³

2015

Nagoya Math. J.

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Consider an elliptic curve defined over an imaginary quadratic field K with good reduction at the primes above p ≥ 5 and has complex multiplication by the full ring of integers OK of K. In this paper, we construct p-adic analogues of the Eisenstein-Kronecker series for such elliptic curve as Coleman functions on the elliptic curve. We then prove p-adic analogues of the first and second Kronecker limit formulas by using the distribution relation of the Kronecker theta function. K coln−m (0, z, n) := E col m,n (z) to highlight the analogy. Then in analogy with Theorem 1.1 (ii), we have the following.Theorem 1.2 (p-adic second Kronecker limit formula). For any prime p ≥ 5 of good reduction, we have the second limit formulawhere log p θ(z) is a certain p-adic analogue of the function log |θ(z)|−|z| 2 /A defined in Definition 5.1 using the reduced theta function θ(z) and the branch of our p-adic logarithm.

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“…Assume just for now that p is a prime for which (32) has good ordinary reduction at p, and suppose that the order of z 0 is prime to p. In [BK1], we defined a two-variable p-adic measure µ z 0 ,0 on Z p × Z p . By substituting η p (t) = exp(λ(t)/Ω p )− 1 into Definition 3.2 of [BK1], we see that the p-adic measure µ z 0 ,0 is defined to satisfy…”

Section: Classical and P-adic Eisenstein-kronecker Numbersmentioning

confidence: 99%

“…Assume now that p is ordinary of the form (p) = pp * in O K , and suppose that z 0 is non-zero of order prime to p. Then for their construction of the two-variable p-adic L-function of the CM elliptic curve (see also [BK1]), Manin-Vishik and Katz constructed a p-adic measure µ z 0 ,0 on Z p × Z p which satisfies…”

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confidence: 99%

“…The definition of Eisenstein-Kronecker numbers for ordinary p also extends to non-zero torsion points z 0 of order prime to p (in other words, the annihilator of z 0 as an element in an O K -module is prime to p). In both constructions, the crucial fact that we use is the main result of [BK1], which states that the generating function for Eisenstein-Kronecker numbers is given by the Kronecker theta function Θ z 0 ,w 0 (z, w).…”

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On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves

Bannai

Kobayashi

Tsuji

2010

Ann. Sci. École Norm. Sup.

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Abstract. In this paper, we give an explicit description of the de Rham and p-adic polylogarithms for elliptic curves using the Kronecker theta function. We prove in particular that when the elliptic curve has complex multiplication and good reduction at p, then the specializations to torsion points of the p-adic elliptic polylogarithm are related to p-adic Eisenstein-Kronecker numbers, proving a p-adic analogue of the result of Beilinson and Levin expressing the complex elliptic polylogarithm in terms of Eisenstein-Kronecker-Lerch series. Our result is valid even if the elliptic curve has supersingular reduction at p.0. Introduction 0.1. Introduction. In the paper [BL], Beilinson and Levin constructed the elliptic polylogarithm, which is an element in absolute Hodge or ℓ-adic cohomology of an elliptic curve minus the identity. This construction is a generalization to the case of elliptic curves of the construction by Beilinson and Deligne of the polylogarithm sheaf on the projective line minus three points. The purpose of this paper is to study the p-adic realization of the elliptic polylogarithm for an elliptic curve with complex multiplication, and to investigate its relation to p-adic L-functions associated to the elliptic curve, even for the case where the elliptic curve has supersingular reduction at the prime p.To achieve our goal, we first describe the de Rham realization of the elliptic polylogarithm for a general elliptic curve defined over a subfield of C. In particular, we explicitly describe the connection of the elliptic polylogarithm using rational functions. This result is of independent interest. Similar results were obtained by Levin and Racinet [LR] Section 5.1.3. A different description for this connection was also given by Besser and Solomon [BS].Using the de Rham realization of the elliptic polylogarithm, we then construct the p-adic realization of the elliptic polylogarithm as a filtered overconvergent F -isocrystal on the elliptic curve minus the identity, when the elliptic curve has complex multiplication and good reduction at a fixed prime p ≥ 5. Our main result Theorem 4.15 is an explicit description of the p-adic Using this description, we calculate the specializations of the p-adic elliptic polylogarithm to torsion points of order prime to p (more precisely, torsion points of order prime to p. See the Overview for details), and prove that the specializations give the p-adic Eisenstein-Kronecker numbers, which are special values of the p-adic distribution interpolating Eisenstein-Kronecker numbers. This result is a generalization of the result of [Ba3], where we have dealt only with the one variable case for an ordinary prime. A similar result concerning the specialization in two-variables was obtained in [BKi], again for ordinary primes, using a very different method. The result of the current paper is valid even when p is supersingular.The p-adic Eisenstein-Kronecker numbers are related to special values of p-adic L-functions which p-adically interpolate special values of Hecke L-f...

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Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers

Abstract: In this paper, we give an overview of our previous paper concerning the investigation of the algebraic and p-adic properties of Eisenstein-Kronecker numbers using Mumford's theory of algebraic theta functions.

Cited by 22 publications

References 32 publications

Eisenstein–kronecker Series via the Poincaré Bundle

Eisenstein–kronecker Series via the Poincaré Bundle

p-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas

On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves

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