Eisenstein–kronecker Series via the Poincaré Bundle

Abstract

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.

Periods of Hilbert Modular forms, Kronecker series and Cohomology

“…A striking result by Zagier [24] states that this rational information of modular forms can be written as a single product of Kronecker series F τ (u, v) which is a Jacobi form. The recent results in [2,21] show that Eisenstein-Kronecker numbers have a rich arithmetic nature, such as a connection with the special Hecke L-function over imaginary quadratic fields and Katz' two-variable p-adic Eisenstein measure. In this paper, we identified the Kronecker series as a "Kuznetsov lifting" of holomorphic Hilbert Eisenstein series over totally real number fields with strict class number 1.…”

The algebraic de Rham realization of the elliptic polylogarithm via the Poincaré bundle

“…The results presented in this paper are part of my PhD thesis at the Universität Regensburg [Spr17]. It is a pleasure to thank my advisor Guido Kings for his guidance during the last years.…”

The Syntomic Realization of the Elliptic Polylogarithm via the Poincaré Bundle

“…The results presented in this paper are part of my Ph.D. thesis [Spr17]. It is a pleasure to thank my advisor Guido Kings for his guidance during the last years.…”

“…The geometry of the Poincaré bundle serves as a substitute for the Kronecker theta function. More precisely, we make use of the purely algebraically defined Kronecker section of the Poincaré bundle, which has been fruitfully applied in [Spr18a] to study algebraic and p-adic properties of Eisenstein-Kronecker series. Since the syntomic realization refines the algebraic de Rham realization, we need a good understanding of the latter.…”

“…Building on previous work of Scheider, we have given there an explicit description of the algebraic de Rham realization for arbitrary families of elliptic curves via the Poincaré bundle. It is again the Poincaré bundle which allows us to relate the syntomic realization to Katz' two variable p-adic Eisenstein measure: Here we build on [Spr18a], where we have constructed Katz' Eisenstein measure using p-adic theta functions associated to the Poincaré bundle.…”