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EISENSTEIN–KRONECKER SERIES VIA THE POINCARÉ BUNDLE

Part of: Discontinuous groups and automorphic forms Arithmetic algebraic geometry

Published online by Cambridge University Press:  30 September 2019

JOHANNES SPRANG Open the ORCID record for JOHANNES SPRANG [Opens in a new window]
JOHANNES SPRANG*
Affiliation:
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany; johannes.sprang@ur.de

Abstract

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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.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 7 , 2019 , e34
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2019

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