Let k = F_q(t) , with q odd. In this article we introduce «definite» (with respect to the infinite place of ) Shimura curves over and establish Hecke module isomorphisms between their Picard groups spaces Drinfeld type «new» forms corresponding level. An important application is a function field analogue Gross formula for central critical values Rankin L -series coming from automorphic cusp type.

The aim of this article is to prove the Siegel-Weil formula over function fields for dual reductive pair (Sp n , O(V )), where Sp symplectic group degree 2n and (V, Q V ) an anisotropic quadratic space with even dimension.This a field analogue Kudla Rallis' result.By formula, theta series identified special value Siegel-Eisenstein on at critical point.

In this paper, we derive a function field version of the Waldspurger formula for central critical values Rankin-Selberg $L$-functions. This states that $L$-values in question can be expressed as "ratio" global toric period integral to product local integrals. Consequently, result provides necessary and sufficient criterion non-vanishing these $L$-values, supports Gross-Prasad conjecture $\mathrm {SO}(3)$ over fields.

Let K be a global function field together with place |$\infty $|⁠, and A the subring of functions regular outside $|⁠. In this paper we present an effective method to evaluate (locally free) class number arbitrary hereditary A-order in definite central simple K-algebra. We also show that any nonprincipal genus for order D can reduced principal another D.

The aim of this paper is to study the central critical value Rankin-type L-functions coming from 'Drinfeld-type' automorphic cusp forms convolved with 'imaginary' quadratic characters. Rankin–Selberg method provides us a very explicit functional equation for these L-functions. When 'root number' in question positive, we derive Gross-type formula over arbitrary global function field. Via theta series constructed definite pure quaternions, then establish Shimura correspondence between...

Fix an integer ℓ≥3. Rikuna introduced a polynomial r(x,t) defined over function field K(t) whose Galois group is cyclic of order ℓ, where K satisfies some mild hypotheses. In this paper we define the family generalized polynomials {r n (x,t)} n≥1 degree ℓ . The r (x,t) are constructed iteratively from r(x,t). We compute groups for odd arbitrary base and give applications to arithmetic dynamical systems.

Let $v$ be a finite place of $\mathbb{F}_q(\theta)$. In this paper, we interpret $v$-adic arithmetic gamma values in terms the crystalline-de Rham periods Carlitz motives with Complex Multiplication, and establish an Ogus-type Chowla-Selberg formula. Furthermore, prove algebraic independence these by determining dimension motivic Galois groups through adaptation refinement existing methods. As consequence, all relations among over $\mathbb{F}_q(\theta)$ can derived from standard functional...

The aim of this paper is to present a function field analogue the classical Kronecker limit formula. We first introduce ``non-holomorphic'' Eisenstein series on Drinfeld half plane, and connect its ``second term'' with Gekeler's discriminant function. One application express Taguchi height rank $2$ modules complex multiplication in terms logarithmic derivative corresponding zeta functions. Moreover, from integral form Rankin-type $L$-function associated two ``Drinfeld-type'' newforms, we...

The aim of this article is to study the derivative ‘incoherent’ Siegel–Eisenstein series on symplectic groups over function fields. Comparing Fourier coefficients incoherent with ‘coherent’ ones, Siegel–Weil formula enables us understand non-singular in question by theta (together a local quantity coming from corresponding Whittaker functions). Restricting special case when quadratic space has dimension 2, we explicitly compute all coefficients, and express terms degree cycles coarse moduli...

Introduction Brandt matrices and definite Shimura curves The basis problem for Drinfeld type automorphic forms Metaplectic Shintani-type correspondence Trace formula of Bibliography Symbols

We establish a general Kronecker limit formula of arbitrary rank over global function fields with Drinfeld period domains playing the role upper-half plane. The Drinfeld-Siegel units come up as equal characteristic modular forms replacing classical $\Delta$. This leads to analytic means deriving Colmez-type for "stable Taguchi height" CM modules having rank. A Lerch-Type "totally real" is also obtained, Heegner cycle on Bruhat-Tits buildings intervene. Also our naturally applied special...

We study the Eisenstein ideal of Drinfeld modular curves small levels, and relation to cuspidal divisor group component groups Jacobians curves. prove that characteristic function field is an prime number when level arbitrary non square-free $\mathbb{F}_q[T]$ not equal a square prime.

Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup Jacobian variety $J_0(\frak{n})$ Drinfeld modular curve $X_0(\frak{n})$. prove that for any prime number $\ell$ not dividing $q(q-1)$, $\ell$-primary part this group coincides with cuspidal divisor class group. further determine structure $q-1$.

In this paper, we derive a function field version of the Waldspurger formula for central critical values Rankin-Selberg L-functions. This states that L-values in question can be expressed as "ratio" global toric period integral to product local integrals. Consequently, result provides necessary and sufficient criterion non-vanishing these L-values, supports Gross-Prasad conjecture $SO(3)$ over fields.