Related papers: Algebraic twists of GL(2) automorphic forms

It is shown that $A:=H_{1,\eta}(G)$, the Sympectic Reflection Algebra, has $T_G$ independent traces, where $T_G$ is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group $G$ generated by the system…

In this paper various analytic techniques are com- bined in order to study the average of a product of a Hecke L- function and a symmetric square L-function at the central point in the weight aspect. The evaluation of the second main term…

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k…

We prove formulas for power moments for point counts of elliptic curves over a finite field $k$ such that the groups of $k$-points of the curves contain a chosen subgroup. These formulas express the moments in terms of traces of Hecke…

Let $G$ and $\tilde G$ be reductive groups over a local field $F$. Let $\eta : \tilde G \to G$ be a $F$-homomorphism with commutative kernel and commutative cokernel. We investigate the pullbacks of irreducible admissible…

Let F be a field with characteristic two. We generalize the second trace form for central simple algebras with odd degree over F. We determine the second trace form and the Arf invariant and Clifford invariant for tensor products of central…

Let $g$ be a Hecke-Maass cusp form on the modular surface ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$, namely an $L^2$-normalised nonconstant Laplacian eigenfunction on ${\rm SL}_2(\mathbb{Z})\backslash\mathbb{H}$ that is additionally a…

For a given l-adic sheaf F on a commutative algebraic group over a finite field k and an integer r we define the r-th local norm L-function of F at a point t in G(k) and prove its rationality. This function gives information on the sum of…

We establish the equivalence of conjectures concerning the pair correlation of zeros of $L$-functions in the Selberg class and the variances of sums of a related class of arithmetic functions over primes in short intervals. This extends the…

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group…

We treat an unbalanced shifted convolution sum of Fourier coefficients of cusp forms. As a consequence, we obtain an upper bound for correlation of three Hecke eigenvalues of holomorphic cusp forms $\sum_{H\leq h\leq…

We introduce and study certain hyperbolic versions of automorphic Lie algebras related to the modular group. Let $\Gamma$ be a finite index subgroup of $\mathrm{SL}(2,\mathbb{Z})$ with an action on a complex simple Lie algebra $\mathfrak…

We prove multiplicity one for vector valued holomorphic Siegel modular forms of weights greater or equal to 3 and the full Siegel modular group and give a trace formula for the action of the Hecke operators T(p) in the regular cases.

We apply some ideas of Bombieri and Garrett to construct natural self-adjoint operators on spaces of automorphic forms whose only possible discrete spectrum is $\lambda_{s}$ for $s$ in a subset of on-line zeros of an $L$-function, appearing…

In this article we investigate the action of (ramified and unramified) Hecke operators on automorphic forms for the function field of the projective line defined over a finite field and for the group GL_2. We first compute the dimension of…

Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a totally imaginary quadratic extension of $F$. We estimate central values of the $\operatorname{GL}_2 \times…

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$.…

Let $F$ be the element $\sum_{n\ \mathit{odd},\ n>0}x^{n^{2}}$ of $Z/2[[x]]$. Set $G=F(x^{5})$, $D=F(x)+F(x^{25})$. For $k>0$, $(k,10)=1$, define $D_{k}$ as follows. $D_{1}=D$, $D_{3}=D^{8}/G$, $D_{7}=D^{2}G$, $D_{9}=D^{4}G$; furthermore…

The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After…

We introduce a natural definition for sums of the form \[ \sum_{\nu=1}^x f(\nu) \] when the number of terms x is a rather arbitrary real or even complex number. The resulting theory includes the known interpolation of the factorial by the…