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

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.