Higher theta series for unitary groups over function fields

Theta Series of Quaternion Algebras over Function Fields

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question.

We prove that all Langlands–Shahidi automorphic L-functions over function fields are rational; after twists by highly ramified characters they become polynomials; and, if $$\pi $$ is a globally generic cuspidal automorphic representation of a split classical group or a unitary group and $$\tau $$ is a cuspidal (unitary) automorphic representation of a general linear group, then $$L(s,\pi \times \tau )$$ is holomorphic for $$\mathfrak {R}(s) > 1$$ and has at most a simple pole at $$s=1$$ . We also prove the holomorphy and non-vanishing of automorphic exterior square, symmetric square and Asai L-functions for $$\mathfrak {R}(s) > 1$$ . Finally, we complete previous results on functoriality for the classical groups over function fields with applications to the Ramanujan Conjecture and Riemann Hypothesis.

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.

We construct analogues of theta series, Eisenstein series and Poincaré series for function fields of one variable over finite fields, and prove their basic properties.

This work is made of two different parts. The first contains results concerning isospectral quadratic forms, and the second is about regular quadratic forms. Two quadratic forms are said to be isospectral if they have the same representation numbers. In this work, we consider binary and ternary definite integral quadratic form defined over the polynomial ring F[t], where F is a finite field of odd characteristic. We prove that the class of such a form is determined by its representation numbers. Equivalently, we prove that there is no nonequivalent definite F[t]-lattices of rank 2 or 3 having the same theta series. A quadratic form is said to be regular (resp. spinor-regular) if it represents any element represented by its genus (resp. by its spinor genus). A form is said to be universal if it represents any integral element. We prove that regular and spinor-regular definite F[t]-lattices must have class number one and we give a characterization of definite universal F[t]-lattices.

LetL be an imaginary quadratic extension of the rational function field $$\mathbb{F}_q (t)$$ . We prove transformation rules for the theta series corresponding to partial zeta functions of the extension $$L/\mathbb{F}_q (t)$$ .

Quotients of theta series as rational functions of j1,4

In analogy with the study of representations of GL 2 n ⁡ ( F ) \operatorname {GL}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) , where F F is a local field, we study representations of U 2 n ⁡ ( F ) \operatorname {U}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) in this paper. (Only quasisplit unitary groups are considered in this paper since they are the only ones which contain Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) .) We prove that there are no cuspidal representations of U 2 n ⁡ ( F ) \operatorname {U}_{2n}(F) distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) for F F a nonarchimedean local field. We also prove the corresponding global theorem that there are no cuspidal automorphic representations of U 2 n ⁡ ( A k ) \operatorname {U}_{2n}(\mathbb {A}_k) with nonzero period integral on Sp 2 n ⁡ ( k ) ∖ Sp 2 n ⁡ ( A k ) \operatorname {Sp}_{2n}(k) \backslash \operatorname {Sp}_{2n}(\mathbb {A}_k) for k k any number field or a function field. We completely classify representations of quasisplit unitary groups in four variables over local and global fields with nontrivial symplectic periods using methods of theta correspondence. We propose a conjectural answer for the classification of all representations of a quasisplit unitary group distinguished by Sp 2 n ⁡ ( F ) \operatorname {Sp}_{2n}(F) .

We calculate the reduced unitary Whitehead groups of skew fields of quotients of noncommutative polynomial rings. We prove a stability theorem in the case where the noncommutative polynomial ring is related to an inner automorphism of the original skew field.

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

Let X be the Fermat curve of degree q+1 over the field k of q elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.

Given a field K equipped with a set of discrete valuations V , we develop a general theory to relate reduction properties of skew-hermitian forms over a quaternion K -algebra Q to quadratic forms over the function field K(Q) obtained via Morita equivalence. Using this we show that if (K,V) satisfies certain conditions, then the number of K -isomorphism classes of the universal coverings of the special unitary groups of quaternionic skew-hermitian forms that have good reduction at all valuations in V is finite and bounded by a value that depends on size of a quotient of the Picard group of V and the size of the kernel and cokernel of residue maps in Galois cohomology of K with finite coefficients. As a corollary we prove a conjecture of Chernousov, Rapinchuk, Rapinchuk for groups of this type.

The graded ring of Hermitian modular forms of degree 2 over [formula omitted

The graded ring of Hermitian modular forms of degree 2 over