On exponential sums for coefficients of general L-functions

We investigate the order of exponential sums involving the coefficients of general [Formula: see text]-functions satisfying a suitable functional equation and give some new estimates, including refining certain results in preceding works [X. Ren and Y. Ye, Resonance and rapid decay of exponential sums of Fourier coefficients of a Maass form for [Formula: see text], Sci. China Math. 58(10) (2015) 2105–2124; Y. Jiang and G. Lü, Oscillations of Fourier coefficients of Hecke–Maass forms and nonlinear exponential functions at primes, Funct. Approx. Comment. Math. 57 (2017) 185–204].

The second moment of symmetric square L-functions over Gaussian integers

We prove a new upper bound on the second moment of Maass form symmetric square L-functions defined over Gaussian integers. Combining this estimate with the recent result of Balog–Biro–Cherubini–Laaksonen, we improve the error term in the prime geodesic theorem for the Picard manifold.

Schneider–Siegel theorem for a family of values of a harmonic weak Maass form at Hecke orbits

Let {j(z)} be the modular j-invariant function. Let τ be an algebraic number in the complex upper half plane {\mathbb{H}}. It was proved by Schneider and Siegel that if τ is not a CM point, i.e., {[\mathbb{Q}(\tau):\mathbb{Q}]\neq 2}, then {j(\tau)} is transcendental. Let f be a harmonic weak Maass form of weight 0 on {\Gamma_{0}(N)}. In this paper, we consider an extension of the results of Schneider and Siegel to a family of values of f on Hecke orbits of τ. For a positive integer m, let {T_{m}} denote the m-th Hecke operator. Suppose that the coefficients of the principal part of f at the cusp {i\infty} are algebraic, and that f has its poles only at cusps equivalent to {i\infty}. We prove, under a mild assumption on f, that, for any fixed τ, if N is a prime such that {N\geq 23} and {N\notin\{23,29,31,41,47,59,71\}}, then {f(T_{m}.\tau)} are transcendental for infinitely many positive integers m prime to N.

Bounds for twisted symmetric square L-functions via half-integral weight periods

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 product L-functions, where $\Psi $ is a fixed Hecke–Maass form on $\operatorname {\mathrm {SL}}_2(\mathbb {Z})$ and $\varphi $ runs over the Hecke–Maass newforms on $\Gamma _0(p)$ of bounded eigenvalue. The proof is via the theta correspondence and analysis of periods of half-integral weight modular forms. This estimate is not expected to be optimal, but the exponent $5/4$ is the strongest obtained to date for a moment problem of this shape. We show that the expected upper bound follows if one assumes the Ramanujan conjecture in both the integral and half-integral weight cases. Under the triple product formula, our result may be understood as a strong level aspect form of quantum ergodicity: for a large prime p, all but very few Hecke–Maass newforms on $\Gamma _0(p) \backslash \mathbb {H}$ of bounded eigenvalue have very uniformly distributed mass after pushforward to $\operatorname {\mathrm {SL}}_2(\mathbb {Z}) \backslash \mathbb {H}$ . Our main result turns out to be closely related to estimates such as $$\begin{align*}\sum_{|n| < p} L(\Psi \otimes \chi_{n p},\tfrac{1}{2}) \ll p, \end{align*}$$ where the sum is over those n for which $n p$ is a fundamental discriminant and $\chi _{n p}$ denotes the corresponding quadratic character. Such estimates improve upon bounds of Duke–Iwaniec.

On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

A SHIFTED CONVOLUTION SUM OF AND THE FOURIER COEFFICIENTS OF HECKE–MAASS FORMS II

Let $d_{3}(n)$ be the divisor function of order three. Let $g$ be a Hecke–Maass form for $\unicode[STIX]{x1D6E4}$ with $\unicode[STIX]{x1D6E5}g=(1/4+t^{2})g$. Suppose that $\unicode[STIX]{x1D706}_{g}(n)$ is the $n$th Hecke eigenvalue of $g$. Using the Voronoi summation formula for $\unicode[STIX]{x1D706}_{g}(n)$ and the Kuznetsov trace formula, we estimate a shifted convolution sum of $d_{3}(n)$ and $\unicode[STIX]{x1D706}_{g}(n)$ and show that $$\begin{eqnarray}\mathop{\sum }_{n\leq x}d_{3}(n)\unicode[STIX]{x1D706}_{g}(n-1)\ll _{t,\unicode[STIX]{x1D700}}x^{8/9+\unicode[STIX]{x1D700}}.\end{eqnarray}$$ This corrects and improves the result of the author [‘Shifted convolution sum of $d_{3}$ and the Fourier coefficients of Hecke–Maass forms’, Bull. Aust. Math. Soc.92 (2015), 195–204].

On an analogue of prime vectors among integer lattice points in ellipsoids for automorphic forms

In this paper, we firstly generalize a result of Friedlander and Iwaniec. By analogy, let [Formula: see text] denote the [Formula: see text]th coefficient of [Formula: see text]-function [Formula: see text] associated to a Hecke–Maass form [Formula: see text] for [Formula: see text] and [Formula: see text] denote the number of representations of [Formula: see text] by the quadratic form [Formula: see text], we investigate the sum with [Formula: see text] [Formula: see text] under a certain assumption.