arXiv subjects

Félicien Comtat

Publications and source records attributed to Félicien Comtat.

7 recordsLinked to original sources

arXiv2025

A weighted vertical Sato-Tate law for Maaß forms on $\rm{GSp}_4$

We prove a weighted Sato-Tate law for the Satake parameters of automorphic forms on $\rm{GSp}_4$ with respect to a fairly general congruence subgroup $H$ whose level tends to infinity. When the level is squarefree we refine our result to the cuspidal spectrum. The ingredients are the $\rm{GSp}_4$ Kuznetsov formula and the explicit calculation of local integrals involved in the Whittaker coefficients of $\rm{GSp}_4$ Eisenstein series. We also discuss how the problem of bounding the continuous spectrum in the level aspect naturally leads to some combinatorial questions involving the double cosets in $P \backslash G / H$, for each parabolic subgroup $P$.

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arXiv2025

An effective version of the Kuznetsov trace formula for GSp(4)

We develop an explicit version of the Kuznetsov trace formula for GSp(4), relating sums of Fourier coefficients to Kloosterman sums. We study the precise analytic behaviour of both the spectral and the arithmetic transforms arising in the Kuznetsov trace formula for GSp(4). We use these results to provide an effective version of the trace formula, and establish various results on the family of Maaß automorphic forms on GSp(4) in the spectral aspect: the Weyl law, a density result on the non-tempered spectrum, large sieve inequalities, bounds on the second moment of the spinor and standard $L$-functions, as well as a statement on the distribution of the low-lying zeros of these $L$-functions, determining the associated types of symmetry.

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arXiv2025

Bounds on Fourier coefficients and global sup-norms for Siegel cusp forms of degree 2

Let $F$ be an $L^2$-normalized Siegel cusp form for $\mathrm{Sp}_4(\mathbb{Z})$ of weight $k$ that is a Hecke eigenform and not a Saito--Kurokawa lift. Assuming the Generalized Riemann Hypothesis, we prove that its Fourier coefficients satisfy the bound $|a(F,S)| \ll_ε\frac{k^{1/4+ε} (4π)^k}{Γ(k)} c(S)^{-\frac12} \det(S)^{\frac{k-1}2+ε}$ where $c(S)$ denotes the gcd of the entries of $S$, and that its global sup-norm satisfies the bound $\|(\det Y)^{\frac{k}2}F\|_\infty \ll_εk^{\frac54+ε}.$ The former result depends on new bounds that we establish for the relevant local integrals appearing in the refined global Gan-Gross-Prasad conjecture (which is now a theorem due to Furusawa and Morimoto) for Bessel periods.

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arXiv2024

Moments of symmetric square L-functions on GL(3)

We give an asymptotic formula with power saving error term for the twisted first moment of symmetric square L-functions on GL(3) in the level aspect. As applications, we obtain non-vanishing results as well as lower bounds of the expected order of magnitude for all even moments, supporting the random matrix model for a unitary ensemble. Besides the GL(3) Kuznetsov formula, the ingredients include detailed local computations at ramified places, including root numbers and orthonormalization of oldforms and Eisenstein series.

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arXiv2022

Sup norms of newforms on $GL_2$ with highly ramified central character

Recently, the problem of bounding the sup norms of $L^2$-normalized cuspidal automorphic newforms $ϕ$ on $\text{GL}_2$ in the level aspect has received much attention. However at the moment strong upper bounds are only available if the central character $χ$ of $ϕ$ is not too highly ramified. In this paper, we establish a uniform upper bound in the level aspect for general $χ$. If the level $N$ is a square, our result reduces to $$\|ϕ\|_\infty \ll N^{\frac14+ε},$$ at least under the Ramanujan Conjecture. In particular, when $χ$ has conductor $N$, this improves upon the previous best known bound $\|ϕ\|_\infty \ll N^{\frac12+ε}$ in this setup (due to Saha [14]) and matches a lower bound due to Templier [17], thus our result is essentially optimal in this case.

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arXiv2021

A relative trace formula approach to the Kuznetsov formula on $GSp_4$

Relative trace formulas play a central role in studying automorphic forms. In this paper, we use a relative trace formula approach to derive a Kuznetsov type formula for the group $GSp_4$. We focus on giving a final formula that is as explicit as possible, and we plan on returning to applications elsewhere.

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arXiv2018

A uniform estimate for the density of rational points on quadrics

This paper is concerned with the density of rational points of bounded height lying on a variety defined by an integral quadratic form Q. In the case of four variables, we give an estimate that does not depend on the coefficients of Q. For more variables, a similar estimate still holds with the restriction that we only count points which do not lie on rational lines.

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