On central critical values of Rankin-type L-functions over global function fields

The aim of this paper is to study the central critical value of Rankin-type L-functions coming from 'Drinfeld-type' automorphic cusp forms convolved with 'imaginary' quadratic characters. Rankin-Selberg method provides us with a very explicit functional equation for these Rankin-type L-functions. When the 'root number' in question is positive, we derive a Gross-type formula over arbitrary global function field. Via the theta series constructed from definite pure quaternions, we then establish a Shimura correspondence between Drinfeld-type forms and metaplectic forms on (SL) over tilde (2). Having this correspondence at hand leads us to an explicit Waldspurger-type formula in this setting.