arXiv subjects

Ramon Nunes

Publications and source records attributed to Ramon Nunes.

5 recordsLinked to original sources

arXiv2026

Regularized spectral expansion of a Rankin--Selberg period and moments

We establish a regularized spectral decomposition of the squared magnitude of a maximal degenerate Eisenstein series as a Schwartz distribution on $\mathrm{GL}_n$. Although the original problem is on $\mathrm{GL}_n$, its spectral components are described entirely by the automorphic spectrum of $\mathrm{GL}_2$. As an application, we prove a partial reciprocity formula relating the second moment of $\mathrm{GL}_n\times\mathrm{GL}_n$ Rankin--Selberg central $L$-values and a mixed moment of $L$-values of $\mathrm{GL}_2$. We also prove, assuming the generalized Ramanujan conjecture, an asymptotic formula for the second moment of the above Rankin--Selberg $L$-functions over a conductor-aspect family with an error term of square-root-cancellation strength.

math.NT↗

arXiv2024

Strong Hybrid Subconvexity for Twisted Selfdual $\mathrm{GL}_3$ $L$-Functions

We prove strong hybrid subconvex bounds simultaneously in the $q$ and $t$ aspects for $L$-functions of selfdual $\mathrm{GL}_3$ cusp forms twisted by primitive Dirichlet characters. We additionally prove analogous hybrid subconvex bounds for central values of certain $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions. The subconvex bounds that we obtain are strong in the sense that, modulo current knowledge on estimates for the second moment of $\mathrm{GL}_3$ $L$-functions, they are the natural limit of the first moment method pioneered by Li and by Blomer. The method of proof relies on an explicit $\mathrm{GL}_3 \times \mathrm{GL}_2 \leftrightsquigarrow \mathrm{GL}_4 \times \mathrm{GL}_1$ spectral reciprocity formula, which relates a $\mathrm{GL}_2$ moment of $\mathrm{GL}_3 \times \mathrm{GL}_2$ Rankin-Selberg $L$-functions to a $\mathrm{GL}_1$ moment of $\mathrm{GL}_4 \times \mathrm{GL}_1$ Rankin-Selberg $L$-functions. A key additional input is a Lindel\"of-on-average upper bound for the second moment of Dirichlet $L$-functions restricted to a coset, which is of independent interest.

math.NT↗

arXiv2023

Moments of $L$-functions via a relative trace formula

We prove an asymptotic formula for the second moment of the $\mathrm{GL}(n)\times\mathrm{GL}(n+1)$ Rankin--Selberg central $L$-values $L(1/2,\Pi\otimes\pi)$, where $\pi$ is a fixed cuspidal representation of $\mathrm{GL}(n)$ that is tempered and unramified at every place, while $\Pi$ varies over a family of automorphic representations of $\mathrm{PGL}(n+1)$ ordered by (archimedean or non-archimedean) conductor. As another application of our method, we prove the existence of infinitely many cuspidal representations $\Pi$ of $\mathrm{PGL}(n+1)$ such that $L(1/2,\Pi\otimes\pi_1)$ and $L(1/2,\Pi\otimes\pi_2)$ do not vanish simultaneously where $\pi_1$ and $\pi_2$ are cuspidal representations of $\mathrm{GL}(n)$ that are unramified and tempered at every place and have trivial central characters.

math.NT↗

arXiv2021

Strong subconvexity for self-dual $\mathrm{GL} (3)$ $L$-functions

In this paper, we prove strong subconvexity bounds for self-dual $\mathrm{GL}(3)$ $L$-functions in the $t$-aspect and for $\mathrm{GL}(3)\times\mathrm{GL}(2)$ $L$-functions in the $\mathrm{GL}(2)$-spectral aspect. The bounds are strong in the sense that they are the natural limit of the moment method pioneered by Xiaoqing Li, modulo current knowledge on estimate for the second moment of $\rm GL(3)$ $L$-functions on the critical line.

math.NT↗

arXiv2021

Spectral reciprocity for $\mathrm{GL}(n)$ and simultaneous non-vanishing of central $L$-values

Let $F$ be a totally real number field and $n\ge 3$. Let $\Pi$ and $\pi$ be cuspidal automorphic representations for $\mathrm{PGL}_{n+1}(F)$ and $\mathrm{PGL}_{n-1}(F)$, respectively, that are unramified and tempered at all finite places. We prove simultaneous non-vanishing of the Rankin--Selberg $L$-values $L(1/2,\Pi\otimes\widetilde{\sigma})$ and $L(1/2,\sigma\otimes\widetilde{\pi})$ for certain sequences of $\sigma$ varying over cuspidal automorphic representations for $\mathrm{PGL}_n(F)$ with conductor tending to infinity in the level aspect and bearing certain local conditions. Along the way, we also prove a reciprocity formula for the average of the product of Rankin--Selberg $L$-functions $L(1/2,\Pi\otimes\widetilde{\sigma})L(1/2,\sigma\otimes\widetilde{\pi})$ over a conductor aspect family of $\sigma$.

math.NT↗