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

Abstract

Following the same framework of the special value results (by Papanikolas and the author) of convolutions of Goss and Pellarin $L$-series attached to Drinfeld modules that take values in Tate algebras, 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.