Tensor products of Drinfeld modules and convolutions of Goss $L$-series

TL;DR

This paper establishes new special value formulas for convolutions of Goss $L$-series associated with Drinfeld modules, using tensor products, class module formulas, and Schur polynomial techniques, with explicit regulator computations.

Contribution

It introduces novel special value results for convolutions of Goss $L$-series via tensor products and class module formulas, extending previous frameworks to new algebraic settings.

Findings

Derived explicit formulas for $L$-series convolutions of Drinfeld modules.

Applied Fang's class module formula to tensor products of Drinfeld modules.

Computed regulators explicitly for tensor, symmetric, and alternating squares.

Abstract

Following the same framework of the special value results of convolutions of Goss and Pellarin $L$-series attached to Drinfeld modules that take values in Tate algebras by Papanikolas and the author, we establish special value results of convolutions of two Goss $L$-series attached to Drinfeld modules that take values in ${\mathbb{F}_q}(\!(\frac{1}{\theta})\!).$ Applying the class module formula of Fang to tensor products of two Drinfeld modules, we provide special value formulas for their $L$-functions. By way of the theory of Schur polynomials these identities take the form of specializations of convolutions of Rankin-Selberg type. Finally, we show an explicit computation of the regulators appearing in Fang's class module formula for tensor products as well as symmetric and alternating squares of Drinfeld modules.