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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References
Dutta Gupta S.: Mean values of L-functions over function fields. J. Number Theory 63, 101–131 (1997)
Gekeler E.-U.: Improper Eisenstein series on Bruhat–Tits trees. Manuscripta Math. 86, 367–391 (1995)
Gekeler E.-U., Reversat M.: Jacobians of Drinfeld modular curves. J. Reine Angew. Math. 476, 27–93 (1996)
Gross, B. H.: Heights and the special values of L-series. In: Kisilevsky, H., Labute, J. (eds.) CMS Conference Proceedings. vol. 7, pp. 116–187 (1987)
Hoffstein J., Rosen M.: Average values of L-series in function fields. J. Reine Angew. Math. 426, 117–150 (1992)
Jacquet, H., Langlands, R.: Automorphic forms on GL(2). Lecture notes in Mathematics, vol. 114. Springer, Berlin (1970)
Kubota, T.: On automorphic functions and the reciprocity law in a number field. Lectures in Mathematics, vol. 21. Kyoto University (1969)
Papikian M.: On the variation of Tate–Shafarevich groups of elliptic curves over hyperelliptic curves. J. Number Theory 115, 249–283 (2005)
Rück H.-G.: Theta series of imaginary quadratic function fields. Manuscripta Math. 88, 387–407 (1995)
Rück H.-G., Tipp U.: Heegner points and L-series of automorphic cusp forms of Drinfeld type. Documenta Math. 5, 365–444 (2000)
Shimura G.: On modular forms of half integral weight. Ann. Math. 97(3), 440–481 (1973)
Tate, J.T.: On the conjectures of Birch and Swinnerton–Dyer and a geometric analog, Séminaire Bourbaki, Exp. No. 306, vol. 9. Soc. Math. France, Paris, pp. 415–440 (1995)
Wei, F.-T.: On arithmetic of curves over function fields. Ph.D. thesis. National Tsing Hua University (2010)
Wei, F.-T., Yu, J.: On theta series from definite quaternion algebras over function fields (preprint)
Wei F.-T., Yu J.: Theta series and function field analogue of Gross formula. Documenta Math. 16, 723–765 (2011)
Weil, A.: Dirichlet series and automorphic forms. Lecture notes in Mathematics, vol. 189. Springer, Berlin (1971)
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This research was partially supported by National Science Council.