Related papers: Special L-values and shtuka functions for Drinfeld…

The purpose of this paper is to give some new Diophantine applications of modularity results. We use the Shimura-Taniyama conjecture to prove effective finiteness results for integral points on moduli schemes of elliptic curves. For several…

We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand-Tsetlin modules with 1-singularity. An explicit construction of such…

In this paper we study the behavior of the function omega of Anderson-Thakur evaluated at the elements of the algebraic closure of the finite field with q elements F_q. Indeed, this function has quite a remarkable relation to explicit class…

Recently we have used the Carlitz exponential map to define a finitely generated submodule of the Carlitz module having the right properties to be a function field analogue of the group of units in a number field. Similarly, we constructed…

We introduce a family of L-series specialising to both L-series associated to certain Dirichlet characters over F_q[T] and to integral values of Carlitz-Goss zeta function associated to F_q[T]. We prove, with the use of the theory of…

We construct special cycles on the moduli stack of unitary shtukas. We prove an identity between (1) the r-th central derivative of non-singular Fourier coefficients of a normalized Siegel--Eisenstein series, and (2) the degree of special…

We investigate Goss $L$-functions associated to Anderson $t$-modules defined by Sinha having complex multiplication by Carlitz cyclotomic fields. We show that these $t$-modules are defined over the cyclotomic field and that their…

We investigate interconnected aspects of hyperderivatives of polynomials over finite fields, q-th powers of polynomials, and specializations of Vandermonde matrices. We construct formulas for Carlitz multiplication coefficients using…

We study a family $\psi^{\lambda}$ of $\mathbb F_q[T]$-Drinfeld modules, which is a natural analog of Legendre elliptic curves. We then find a surprising recurrence giving the corresponding Deuring polynomial $H_{p(T)}(\lambda)$…

We evaluate a rigid analytical analogue of the Beilinson-Bloch-Deligne regulator on certain explicit elements in the K_2 of Drinfeld modular curves, constructed from analogues of modular units, and relate its value to special values of…

We study the special value at 2 of L-functions of modular forms of weight 2 on congruence subgroups of the modular group. We prove an explicit version of Beilinson's theorem for the modular curve X_1(N). When N is prime, we deduce that the…

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let…

We prove an algebraicity result for the central critical value of certain Rankin-Selberg L-functions for GL(n) x GL(n-1). This is a generalization and refinement of some results of Harder, Kazhdan-Mazur-Schmidt, Mahnkopf, and…

We establish several formulas relating periods of modular forms on quaternion algebras over number fields to special values of L-functions. Our main inputs are the cohomological techniques for working with periods introduced in [Mol21],…

In this paper we obtain a weighted average formula for special values of $L$-functions attached to normalized elliptic modular forms of weight $k$ and full level. These results are obtained by studying the pullback of a Siegel Eisenstein…

Feng-Yun-Zhang have proved a function field analogue of the arithmetic Siegel-Weil formula, relating special cycles on moduli spaces of unitary shtukas to higher derivatives of Eisenstein series. We prove an extension of this formula in a…

Inspired by Lehmer's conjecture on the nonvanishing of the Ramanujan $\tau$-function, one may ask whether an odd integer $\alpha$ can be equal to $\tau(n)$ or any coefficient of a newform $f(z)$. Balakrishnan, Craig, Ono, and Tsai used the…

We introduce and study a natural class of Anderson t- modules, called triangular t-modules, characterized by having Drinfeld modules as their $\tau$-composition factors. They form a homologically meaningful generalization of Drinfeld…

For motives associated with Fermat curves, there are elements in motivic cohomology whose regulators are written in terms of special values of generalized hypergeometric functions. Using them, we verify the Beilinson conjecture numerically…

In \cite{FGHP}, the first author and his collaborators proved an equivariant Tamagawa number formula for the special value at $s=0$ of a Goss--type $L$--function, equivariant with respect to a Galois group $G$, and associated to a Drinfeld…