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.
Highlights
The classical Eisenstein–Kronecker series are defined for a lattice Γ = ω1Z +
Building on the work of Bannai–Kobayashi, we provide in the first part of the paper a purely algebraic construction of Eisenstein–Kronecker series via the Poincaré bundle on the universal vectorial biextension
In the second part of the paper, we provide a new approach to the p-adic interpolation of Eisenstein–Kronecker series
Summary
Ω2Z ⊆ C, s, t ∈ (1/N )Γ and integers r − 2 > k 0 by the absolutely convergent series ek∗,r A classical result of Katz gives a purely algebraic approach towards Eisenstein– Kronecker series by giving an algebraic interpretation of the Maaß–Shimura operator on the modular curve in terms of the Gauß–Manin connection His construction has been one of the main sources for studying systematically the algebraic and p-adic properties of real-analytic Eisenstein series. Norman’s theory of p-adic theta functions allows us to associate a p-adic theta function to the Kronecker section for any elliptic curve with ordinary reduction over a p-adic base Applying this to the universal trivialized elliptic curve gives us a two-variable power series Dθ(a,b)(S, T ) over the ring of generalized p-adic modular forms. Let us refer to Theorem 8.1 and Corollary 8.2 for details This construction does give a more conceptional approach towards the p-adic Eisenstein measure, it provides a direct bridge between p-adic theta functions and padic modular forms.
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