The Syntomic Realization of the Elliptic Polylogarithm via the Poincaré Bundle

Sprang, Johannes

doi:10.4171/dm/700

Cited by 2 publications

References 15 publications

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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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Eisenstein–kronecker Series via the Poincaré Bundle

Sprang

2019

Forum of Mathematics, Sigma

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Hyodo-Kato theory with syntomic coefficients

Yamada¹

2020

Preprint

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The purpose of this article is to establish theories concerning p-adic analogues of Hodge cohomology and Deligne-Beilinson cohomology with coefficients in variations of mixed Hodge structures. We first study log overconvergent F -isocrystals as coefficients of Hyodo-Kato cohomology. In particular, we prove a rigidity property of Hain-Zucker type for mixed log overconvergent F -isocrystals. In the latter half of the article, we give a new definition of syntomic coefficients as coefficients of p-adic Hodge cohomology and syntomic cohomology, and prove some fundamental properties concerning base change and admissibility. In the study of syntomic coefficients, the rigid analytic Hyodo-Kato map which for the trivial coefficient was studied by Ertl and the author plays a key role. K (ϕ, N )), and there exists a canonical isomorphism). Because of Theorem 0.3 (3), we may regard the syntomic cohomology as an absolute p-adic Hodge cohomology (a p-adic analogue of the Deligne-Beilinson cohomology).Note that we suppose the properness of X in order to consider Hodge filtrations of the de Rham cohomology, but admit a log structure associated to a horizontal divisor. Therefore we may deal with compactifiable cases.One can also show that, if X is smooth and (E , Φ) is induced from a non-logarithmic overconvergent F -isocrystal, then Ψ π,log is independent of π and log, and coincides with the base change map of the non-logarithmic rigid cohomology (Proposition 8.9). Therefore our theory of syntomic coefficients is compatible with the construction in [2] of the syntomic coefficients of Besser's rigid syntomic cohomology for smooth schemes.Our definitions of syntomic coefficients and the Hyodo-Kato map are useful for direct computation. For example, one may extend the methods of p-adic polylogarithms to suitable group schemes with semistable reduction. In the case of good reduction, the p-adic polylogarithms were constructed as classes of the non-logarithmic version of (rigid) syntomic cohomology with syntomic coefficients, and described explicitly for some special cases [1,3,4,40]. In the case of semistable reduction, the p-adic polylogarithms should be defined as classes of the logarithmic (rigid) syntomic cohomology with syntomic coefficients which we define in §10 of this article. Since our construction is simple and explicit, one may calculate them as Čech-de Rham coycles on p-adic analytic spaces.

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The Syntomic Realization of the Elliptic Polylogarithm via the Poincaré Bundle

Cited by 2 publications

References 15 publications

Eisenstein–kronecker Series via the Poincaré Bundle

Eisenstein–kronecker Series via the Poincaré Bundle

Hyodo-Kato theory with syntomic coefficients

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