• Corpus ID: 119653545

The twisted second moment of modular half integral weight $L$--functions.

@article{Dunn2019TheTS,
  title={The twisted second moment of modular half integral weight \$L\$--functions.},
  author={Alexander Dunn and Alexandru Zaharescu},
  journal={arXiv: Number Theory},
  year={2019}
}
Given a half-integral weight holomorphic Kohnen newform $f$ on $\Gamma_0(4)$, we prove an asymptotic formula for large primes $p$ with power saving error term for \begin{equation*} \sideset{}{^*} \sum_{\chi \hspace{-0.15cm} \pmod{p}} \big | L(1/2,f,\chi) \big |^2. \end{equation*} Our result is unconditional, it does not rely on the Ramanujan-Petersson conjecture for the form $f$. There are two main inputs. The first is a careful spectral analysis of a highly unbalanced shifted convolution… 

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References

SHOWING 1-10 OF 62 REFERENCES

The Second Moment of Twisted Modular L-Functions

AbstractWe prove an asymptotic formula with a power saving error term for the (pure or mixed) second moment $$\underset{\chi \bmod {q}}{\left. \sum \right.^{\ast}} L(1/2, f_1\otimes \chi)

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

Kuznetsov's Trace Formula and the Hecke Eigenvalues of Maass Forms

Introduction Preliminaries Bi-$K_\infty$-invariant functions on $\operatorname{GL}_2(\mathbf{R})$ Maass cusp forms Eisenstein series The kernel of $R(f)$ A Fourier trace formula for

Second Moments and simultaneous non-vanishing of GL(2) automorphic L-series

We obtain a second moment formula for the L-series of holomorphic cusp forms, averaged over twists by Dirichlet characters modulo a fixed conductor Q. The estimate obtained has no restrictions on Q,

Non-vanishing of L-functions attached to automorphic representations of GL (2) over Q.

If / is a new-form of weight 2, trivial character φ and with rational Fourier coefficients (n), Shimura [Shi.l] attached to it an elliptic curve £}, defined over Q, with conductor 7V (in the sense of

The second moment of Dirichlet twists of Hecke L-functions

Fix a Hecke cusp form $f$, and consider the $L$-function of $f$ twisted by a primitive Dirichlet character. As we range over all primitive characters of a large modulus $q$, what is the average

The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros

We describe a number of results and techniques concerning the non-vanishing of automorphic L-functions at s = ½. In particular we show that as N → ∞ at least 50% of the values L(½, f), with f varying

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 ,

On the Fourth Power Moment of the Riemann Zeta-Function

Abstract The function E 2 ( R ) is used to denote the error term in the asymptotic formula for the fourth power moment of the Riemann zeta-function on the half-line. In this paper we prove several

Quantum unique ergodicity for half-integral weight automorphic forms

We investigate the analogue of the Quantum Unique Ergodicity (QUE) conjecture for half-integral weight automorphic forms. Assuming the Generalized Riemann Hypothesis (GRH) we establish QUE for both
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