Theta series and function field analogue of Gross formula
The authors develop a theory of definite quaternion algebras over the rational function field \(\mathbb{F}_q(t)\) for \(q\) odd. For the product \(N_0\) of finite ramified primes, a definite Shimura curve \(X_{N_0}\) is introduced and it is proved that is a finite union of genus zero curves. Special points on \(X_{N_0}\) (Gross points) are defined, which are associated to orders in imaginary quadratic extensions of \( \mathbb{F}_q(t)\). After expressing Hecke correspondences in term of Brandt matrices, certain theta series are introduced. The main theorem of the article concerns the existence of a map \[ \Phi: \mathrm{Pic}(X) \times \mathrm{Pic}^\vee(X) \rightarrow M^{\mathrm{new}}(\Gamma_0(N_0)), \] such that for all monic polynomials \(m\in \mathbb{F}_q[t]\) the following equation holds: \[ T_m\Phi(e,e')=\Phi(t_m e,e')=\Phi(e,t_m e'), \] where \(t_m\) are Hecke correspondences on \(X\), \(T_m\) are Hecke correspondences on \(M^{\mathrm{new}}(\Gamma_0(N_0))\), the space of Drinfeld type ``new'' forms for \(\Gamma_0(N_0)\). For certain quadratic extensions \(K=\mathbb{F}_q[t](\sqrt{D})\) of \(\mathbb{F_q}\) automorphic forms \(g_\mathcal{A}\) corresponding to ideal classes \(\mathcal{A}\) of \(\mathbb{F}_q[t][\sqrt{D}]\) are explicitly constructed. The central critical value of the Rankin product \(L\)-series \(\Lambda(f,\mathcal{A},s)\) associated to a Drinfeld type new form \(f\), is expressed in terms of the Petersson inner product of \(f\) and \(g_\mathcal{A}\). If \(f\) is a normalised Hecke eigenform and \(\chi\) is a character of \(\mathrm{Pic}(\mathbb{F}_q[t][\sqrt{D}])\), the twisted critical value \(\Lambda(f,\chi,0)\) is expressed explicitly in terms of the Gross height of a special divisor class \(e_{f,\chi}\) of the definite Shimura curve \(X_{N_0}\). This is an analogue to Gross formula in the case studied by the authors. Finally, application to \(L\)-series of certain elliptic curves are given.
scientific article; zbMATH DE number 5987893