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

@article{Templier2011NonsplitSO, title={Non-split sums of coefficients of GL(2)-automorphic forms}, author={Nicolas Templier and Jacob Tsimerman}, journal={Israel Journal of Mathematics}, year={2011}, volume={195}, pages={677-723} }

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…

## 16 Citations

Summing Hecke eigenvalues over polynomials

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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

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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.…

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- MathematicsTransactions of the American Mathematical Society
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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.…

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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…

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- 2019

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

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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

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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…

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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…

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- MathematicsForum of Mathematics, Sigma
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Abstract We establish the first moment bound
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Integral presentations of the shifted convolution problem and subconvexity estimates for $\operatorname{GL}_n$-automorphic $L$-functions.

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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…

## References

SHOWING 1-10 OF 42 REFERENCES

A Burgess-like subconvex bound for twisted L-functions

- Mathematics
- 2007

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

- Mathematics
- 2008

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

- Mathematics
- 2009

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

- MathematicsCompositio Mathematica
- 2011

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

- Mathematics
- 2007

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

- Mathematics
- 1973

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

- Mathematics
- 2008

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

- Mathematics
- 1988

(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

- Mathematics
- 2009

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

- Mathematics
- 2001

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…