Gras-Type Conjectures Fattitle for Function Fields

Based on results obtained in [15], we construct groups of special S-units for function fields of characteristic p>0, and show that they satisfy Gras-type Conjectures. We use these results in order to give a new proof of Chinburg's ₃-Conjecture on the Galois module structure of the group of S-units, for cyclic extensions of prime degree of function fields.     

Selberg＇s Normal Density Theorem for Automorphic L-Functions for GLm

Let π be an irreducible unitary cuspidal representation of GLm（AQ） with m ≥ 2, and L（s, Tr） the L-function attached to π. Under the Generalized Riemann Hypothesis for L（s,π）, we estimate the normal density of primes in short intervals for the automorphic L-function L（s, π）. Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.     

The local period integrals and essential vectors

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of GL_n over p-adic fields. In each case, we show that the integrals achieve local formal L-functions defined by Langlands parameters, when the test vector is associated to the new form. We give the relation between local periods involving essential Whittaker functions and special values of formal L-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also served as standard nonzero distinguished forms.