Abstract
We propose a concept of half integral weight in the global function field context, and construct natural families of functions with given weight. An analogue of Shimura correspondence (between weight 2 functions and weight \({\frac{3}{2}}\) functions) via theta series from “definite” quaternion algebras over function fields is then established. From the study of Fourier coefficients of these theta series, we arrive at a Waldspurger-type formula. This formula is then applied to L-series coming from elliptic curves over function fields.
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