Hecke $L$-series for Sinha modules

TL;DR

This paper explores Goss $L$-functions linked to Sinha modules with complex multiplication, demonstrating their relation to Hecke $L$-series and expressing special values via Thakur's $ ext{Gamma}$-function.

Contribution

It establishes that Sinha modules are defined over cyclotomic fields and their $L$-functions factor into Hecke $L$-series, connecting special values to Thakur's $ ext{Gamma}$-function.

Findings

Sinha modules are defined over cyclotomic fields.

Their $L$-functions decompose into Hecke $L$-series.

Special values relate to Thakur's geometric $ ext{Gamma}$-function.

Abstract

We investigate Goss $L$-functions associated to Anderson $t$-modules defined by Sinha having complex multiplication by Carlitz cyclotomic fields. We show that these $t$-modules are defined over the cyclotomic field and that their $L$-functions are products of Hecke $L$-series for Anderson's Hecke character defined via Coleman functions. Applying identities of Fang and Taelman, we prove that special values of these $L$-functions are expressible in terms of products of values of Thakur's geometric $\Gamma$-function.