Shtukas and the Taylor expansion of $L$-functions (II)

Shtukas and the Taylor expansion of $L$-functions

We define the Heegner--Drinfeld cycle on the moduli stack of Drinfeld Shtukas of rank two with $r$-modifications for an even integer $r$. We prove an identity between (1) The $r$-th central derivative of the quadratic base change $L$-function associated to an everywhere unramified cuspidal automorphic representation $π$ of $PGL_{2}$; (2) The self-intersection number of the $π$-isotypic component of the Heegner--Drinfeld cycle. This identity can be viewed as a function-field analog of the Waldspurger and Gross--Zagier formula for higher derivatives of $L$-functions.

Congruences modulo primes of the Romik sequence related to the Taylor expansion of the Jacobi theta constant θ_3

Recently, Romik determined in [9] the Taylor expansion of the Jacobi theta constant θ_3, around the point x = 1. He discovered a new integer sequence, (d(n))_0^\infty=1, 1, -1, 51, 849, -26199, \dots, from which the Taylor coefficients are built, and conjectured that the numbers d(n) satisfy certain congruences modulo various primes. In this paper, we prove some of these conjectures, for example that d(n)\equiv (-1)^{n+1}(mod 5) for all n\geq 1,and that for any prime p\equiv 3 (mod 4), d(n) vanishes modulo p for all large enough n.

关于椭圆曲线复L函数泰勒展开系数的研究

On the coefficients of the Taylor expansion of $L$-functions of elliptic curves

本文研究了椭圆曲线复L函数泰勒展开系数，在GRH下证明了当判别式足够大时，二次扭家族中系数非零。无条件地，通过模形式L函数二次扭中心值导数的矩结果，获得非零系数的下界。

Taylor expansions of Jacobi forms and linear relations among theta series

We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated…

The Taylor coefficients of the Jacobi theta constant $θ_3$

We study the Taylor expansion around the point $x=1$ of a classical modular form, the Jacobi theta constant $θ_3$. This leads naturally to a new sequence $(d(n))_{n=0}^\infty=1,1,-1,51,849,-26199,\ldots$ of integers, which arise as the Taylor coefficients in the expansion of a related "centered" version of $θ_3$. We prove several results about the numbers $d(n)$ and conjecture that they satisfy the congruence $d(n)\equiv (-1)^{n-1}\ (\textrm{mod }5)$ and other similar congruence relations.

Periodicities for Taylor coefficients of half-integral weight modular forms

Congruences of Fourier coefficients of modular forms have long been an object of central study. By comparison, the arithmetic of other expansions of modular forms, in particular Taylor expansions around points in the upper-half plane, has been much less studied. Recently, Romik made a conjecture about the periodicity of coefficients around $τ=i$ of the classical Jacobi theta function. Here, we prove this conjecture and generalize the phenomenon observed by Romik to a general class of modular forms of half-integral weight.