Special L-values and shtuka functions for Drinfeld modules on elliptic curves

TL;DR

This paper develops a detailed theory of rank 1 Drinfeld modules over elliptic curves, deriving explicit formulas for fundamental periods and special L-series values, extending the analogy of the Carlitz module.

Contribution

It introduces a parallel framework for elliptic curve-based Drinfeld modules, including explicit shtuka functions and new identities for L-series values.

Findings

Product formula for the fundamental period of the Drinfeld module

Identities for deformations of reciprocal sums

Proof of special value formulas for Pellarin L-series

Abstract

We make a detailed account of sign-normalized rank 1 Drinfeld A-modules, for A the coordinate ring of an elliptic curve over a finite field, in order to provide a parallel theory to the Carlitz module for F_q[t]. Using precise formulas for the shtuka function for A, we obtain a product formula for the fundamental period of the Drinfeld module. Using the shtuka function we find identities for deformations of reciprocal sums and as a result prove special value formulas for Pellarin L-series in terms of an Anderson-Thakur function. We also give a new proof of a log-algebraicity theorem of Anderson.