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Alexandre de Faveri

Publications and source records attributed to Alexandre de Faveri.

8 recordsLinked to original sources
arXiv2026

Non-orthogonality of the cubic and quartic large sieves via Rankin-Selberg

We show unconditionally that the cubic and quartic large sieves are not perfectly orthogonal. The main obstruction to perfect orthogonality comes from the bias exhibited by Gauss sums. Our proof requires two main inputs: a Lindel\"{o}f-on-average upper bound for the second moment of Kubota's Dirichlet series, and a tight average lower bound for the Fourier coefficients of a certain Rankin-Selberg convolution of metaplectic theta functions. The latter input is particularly important in the quartic case, where much less is known about Fourier coefficients of metaplectic theta functions. To establish both of these inputs, we adapt a Rankin-Selberg regularization method due to Zagier (1981). In addition to the cubic and quartic cases considered in this paper, we expect that the family of Hecke characters of each fixed order $n \geq 3$ over a number field $K \supset \mathbb{Q}(\zeta_n)$ is not perfectly orthogonal. We provide a precise conjecture for the operator norm of these ensembles for each $n$.

math.NT↗
arXiv2026

Non-vanishing for quartic Hecke $L$-functions and ranks of elliptic curves

We show that a positive proportion of Hecke $L$-functions attached to the quartic residue symbols $\big( \frac{\cdot}{q} \big)_4$ for squarefree $q \in \mathbb{Z}[i]$ do not vanish at the central point. Our method also extends to the Hecke characters associated to quartic twists of the congruent number curve $E : y^2 = x^3 - x$. In particular, we prove that the elliptic curve $E^{(q)} : y^2 = x^3 - qx$ has Mordell-Weil rank $0$ over $\mathbb{Q}(i)$ for a positive proportion of squarefree $q \in \mathbb{Z}[i]$ ordered by norm.

math.NT↗
arXiv2026

Mass equidistribution for lifts on hyperbolic $4$-manifolds

This paper is part of a broader project to establish the quantum unique ergodicity conjecture for Hecke-Maass forms on arithmetic hyperbolic $4$-manifolds. The conjecture is known in dimensions $2$ and $3$, but dimensions $4$ and higher present a new difficulty: ruling out concentration along certain homogeneous submanifolds whose stabilizers are non-tempered. We overcome it here for the Pitale lifts, a family of non-holomorphic analogues of Saito-Kurokawa lifts. Our main new ingredient is a construction of an amplifier with exceptional geometric properties. To the best of our knowledge, this is the first successful use of amplification to rule out concentration along a non-tempered subgroup.

math.NT↗
arXiv2024

Non-vanishing for cubic Hecke $L$-functions

Let $\omega$ be a primitive cubic root of unity. We study the non-vanishing problem for the family of Hecke $L$-functions associated to primitive cubic characters defined over the Eisenstein quadratic number field $\mathbb{Q}(\omega)$. We prove unconditionally that a positive proportion of Hecke $L$-functions associated to the cubic residue symbols $\chi_q$ with $q \in \mathbb{Z}[\omega]$ squarefree and $q \equiv 1 \pmod{9}$ do not vanish at the central point. Our proof goes through the method of first and second mollified moments. The principal new contribution of this paper is the asymptotic evaluation of the mollified second moment with power saving error term. No asymptotic formula for the mollified second moment of a cubic family was known (even over function fields) prior to the writing of this paper. Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindel\"{o}f-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the (unitary) family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet $L$-functions in the literature for which unconditional results are known: the symplectic family of quadratic characters and the unitary family of all Dirichlet characters $\chi \pmod{q}$. Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

math.NT↗
arXiv2022

The variance of closed geodesics in balls and annuli on the modular surface

We asymptotically estimate the variance for the distribution of closed geodesics in small random balls or annuli on the modular surface $Γ\backslash\mathbb{H}$. A probabilistic model in which closed geodesics are modeled using random geodesic segments is proposed, and we rigorously analyze this model using mixing of the geodesic flow in $Γ\backslash\mathbb{H}$. This leads to a conjecture for the asymptotic behavior of the variance, which unlike in previously explored cases is not equal to the expected value. We prove this conjecture for small balls and annuli, resolving a question left open by Humphries and Radziwill.

math.NT↗
arXiv2021

Simple zeros of $\mathrm{GL}(2)$ $L$-functions

Let $f \in S_k(\Gamma_1(N))$ be a primitive holomorphic form of arbitrary weight $k$ and level $N$. We show that the completed $L$-function of $f$ has $\Omega\left(T^\delta\right)$ simple zeros with imaginary part in $\left[-T, T\right]$, for any $\delta < \frac{2}{27}$. This is the first power bound in this problem for $f$ of non-trivial level, where previously the best results were $\Omega(\log\log\log{T})$ for $N$ odd, due to Booker, Milinovich, and Ng, and infinitely many simple zeros for $N$ even, due to Booker. In addition, for $f$ of trivial level ($N=1$), we also improve an old result of Conrey and Ghosh on the number of simple zeros.

math.NT↗
arXiv2020

Möbius disjointness for $C^{1 + \varepsilon}$ skew products

We show that for $\varepsilon > 0$, every $C^{1 + \varepsilon}$ skew product on $\mathbb{T}^2$ over a rotation of $\mathbb{T}^1$ satisfies Sarnak's conjecture. This is an improvement of earlier results of Kulaga-Przymus-Lemańczyk, Huang-Wang-Ye, and Kanigowski-Lemańczyk-Radziwill.

math.DS↗