Non-split sums of coefficients of GL(2)-automorphic forms

  title={Non-split sums of coefficients of GL(2)-automorphic forms},
  author={Nicolas Templier and Jacob Tsimerman},
  journal={Israel Journal of Mathematics},
Given a cuspidal automorphic form π on GL2, we study smoothed sums of the form $$\sum\nolimits_n {{a_\pi }({n^2} + d)V({n \over x})} $$. The error term we get is sharp in that it is uniform in both d and Y and depends directly on bounds towards Ramanujan for forms of half-integral weight and Selberg eigenvalue conjecture. Moreover, we identify (at least in the case where the level is square-free) the main term as a simple factor times the residue as s = 1 of the symmetric square L-function L(s… 
Summing Hecke eigenvalues over polynomials
In this paper we estimate sums of the form ∑ n≤X |aSymm π(|f(n)|)|, for symmetric power lifts of automorphic representations π attached to holomorphic forms and polynomials f(x) ∈ Z[x] of arbitrary
Averages of Hecke eigenvalues over thin sequences
Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level.
Counting integer points on quadrics with arithmetic weights
  • V. Kumaraswamy
  • Mathematics
    Transactions of the American Mathematical Society
  • 2020
Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $\lambda(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level.
A spectral proof of class number one
  • M. Watkins
  • Mathematics
    Mathematische Zeitschrift
  • 2018
We continue our previous work on the subject of re-proving the Heegner-Baker-Stark theorem, giving another effective resolution of this conjecture of Gauss, namely there are exactly 9 imaginary
Class group twists and Galois averages of $\operatorname{GL}_n$-automorphic $L$-functions
Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We derive nonvanishing estimates for the finite parts of the $L$-functions of irreducible cuspidal
Quadratic Hecke Sums and Mass Equidistribution
We consider the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke
Effective equidistribution of shears and applications
A “shear” is a unipotent translate of a cuspidal geodesic ray in the quotient of the hyperbolic plane by a non-uniform discrete subgroup of $${\text {PSL}}(2,\mathbb {R})$$PSL(2,R), possibly of
Rankin-Selberg L-functions in cyclotomic towers, III
Let $\pi$ be a cuspidal automorphic representation of $\operatorname{GL}_2$ over a totally real number field $F$. Let $K$ be a quadratic extension of $F$. Fix a prime ideal of $F$, and consider the
Bounds for twisted symmetric square L-functions via half-integral weight periods
Abstract We establish the first moment bound $$\begin{align*}\sum_{\varphi} L(\varphi \otimes \varphi \otimes \Psi, \tfrac{1}{2}) \ll_\varepsilon p^{5/4+\varepsilon} \end{align*}$$ for triple
Integral presentations of the shifted convolution problem and subconvexity estimates for $\operatorname{GL}_n$-automorphic $L$-functions.
Fix $n \geq 2$ an integer, and let $F$ be a totally real number field. We reduce the shifted convolution problem for $L$-function coefficients of $\operatorname{GL}_n({\bf{A}}_F)$-automorphic forms


A Burgess-like subconvex bound for twisted L-functions
Abstract Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, χ a primitive character of conductor q, and s a point on the critical line ℜs = ½. It is proved that ,
Sums of Hecke Eigenvalues over Values of Quadratic Polynomials
Let be a cusp form for Γ 0 (N), weight k ≥ 4, and character χ. Let be a quadratic polynomial. It is shown thatfor some constant c = c(f, q). The constant vanishes in many (but not all) cases, for
A non-split sum of coefficients of modular forms
We shall introduce and study certain truncated sums of Hecke eigenvalues of $GL_2$-automorphic forms along quadratic polynomials. A power saving estimate is established and new applications to
Serre weights for quaternion algebras
Abstract We study the possible weights of an irreducible two-dimensional mod p representation of ${\rm Gal}(\overline {F}/F)$ which is modular in the sense that it comes from an automorphic form on a
The spectral decomposition of shifted convolution sums
Let pi(1), pi(2)) be cuspidal automorphic representations of PGL(2)(R) Qf conductor 1 and Hecke eigenvalues lambda(pi 1,2) (n) and let h > 0 be an integer. For any smooth compactly supported weight
Modular Forms of Half Integral Weight
The forms to be discussed are those with the automorphic factor (cz + d)k/2 with a positive odd integer k. The theta function $$ \theta \left( z \right) = \sum\nolimits_{n = - \infty }^\infty
Hybrid bounds for twisted L-functions
Abstract The aim of this paper is to derive bounds on the critical line ℜs = 1/2 for L-functions attached to twists f ⊗ χ of a primitive cusp form f of level N and a primitive character modulo q that
Hyperbolic distribution problems and half-integral weight Maass forms
(Actually n ~ is replaced by d(n)log ~ 2n where d(n) is the divisor function.) A striking application of(1.2) is to give the uniform distribution of certain lattice points in Z 3 on a sphere centered
Twisted L-Functions Over Number Fields and Hilbert’s Eleventh Problem
Let K be a totally real number field, π an irreducible cuspidal representation of $${{\rm GL}_{2}(K){\backslash}{\rm GL}_{2}(\mathbb{A}K)}$$ with unitary central character, and χ a Hecke character of
An additive problem in the Fourier coefficients of cusp forms
Abstract. We establish an estimate on sums of shifted products of Fourier coefficients coming from holomorphic or Maass cusp forms of arbitrary level and nebentypus. These sums are analogous to the