arXiv subjects
Publications and source records attributed to Philippe Michel.
We study bounds for correlation sums of automorphic coefficients on $\mathrm{GL}_{3,\mathbb{Q}}$ with trace functions of composite moduli. This is a sequel to our previous works with E. Kowalski and W. Sawin.
Generalizing our previous work on ``toroidal averages'', we study the average of special values of $L$-functions of the form $L(1/2,\chi^a)L(1/2,\chi^b)L(1/2,\chi^c)$ for integers $a$, $b$ and $c$, where $\chi$ varies over Dirichlet characters of a given prime modulus. We highlight connections with estimates for bilinear forms of trace functions and with bounds for the number of solutions of monoidal equations in three variables in small boxes over finite fields.
We obtain non-trivial bounds for bilinear sums of trace functions below the P\'olya-Vinogradov range assuming only that the geometric monodromy group of the underlying ell-adic sheaf satisfies certain simple structural properties, in contrast to previous works which handled only special cases of Kloosterman and hypergeometric sheaves. Our approach builds on a general "soft" stratification theorem for sums of products of trace functions, based on an idea of Junyan Xu, combined with a new robust version of the Goursat-Kolchin-Ribet criterion.
Let $f$ be a non-CM elliptic newform without a quadratic inner twist, $p$ an odd prime and $\chi$ a Dirichlet character of $p$-power order and sufficiently large $p$-power conductor. We show that the compositum $\mathbb{Q}_{f}(\chi)$ of the Hecke fields associated to $f$ and $\chi$ is generated by the square of the absolute value of the corresponding central $L$-value $L^{\rm alg}(1/2, f \otimes \chi)$ over $\mathbb{Q}(\mu_p)$. The proof is based among other things on techniques used for the recent resolution of unipotent mixing conjecture by the first and third named authors.
Fix a Dirichlet character $\chi$ and a cuspidal GL$(2)$ eigenform $\phi$ with relatively prime conductors. Then we show that there are infinitely many cusp forms $\pi$ on GL$(3)$ such that $L(1/2, \pi \times \chi)$ and $L(1/2, \pi \times \phi)$ are simultaneously non-zero. We achieve this by use of Jacquet's Relative Trace Formula. We derive an expression of the average over the GL$(3)$ cuspidal spectrum as a sum of a non-zero main term and two subsidiary terms which are forced to be zero for large enough level by use of a suitable test function. This modest article is dedicated to the memory of Harish Chandra, on the occasion of his hundredth birthday.
In this paper we calculate the asymptotics of the second moment of the Bessel periods associated to certain holomorphic cuspidal representations $(\pi, \pi')$ of $U(2,1) \times U(1,1)$ of regular infinity type (averaged over $\pi$). Using these, we obtain quantitative non-vanishing results for the Rankin-Selberg central $L$-values $L(1/2, \pi \times \pi')$, which are of degree twelve over $\mathbb{Q}$, with concomitant difficulty in applying standard methods, especially since we are in a `conductor dropping' situation. We use the relative trace formula, and the orbital integrals are evaluated rather than compared with others. Besides their intrinsic interest, non-vanishing of these critical values also lead, by known results, to deducing certain associated Selmer groups have rank zero.
We show that shifted pairs of discrete or continuous low-lying horocycles equidistribute in the product space of two modular curves.
We study bounds for algebraic twists sums of automorphic coefficients by trace functions of composite moduli.
Let $(\lambda_f(n))_{n\geq 1}$ be the Hecke eigenvalues of either a holomorphic Hecke eigencuspform or a Hecke-Maass cusp form $f$. We prove that, for any fixed $\eta>0$, under the Ramanujan-Petersson conjecture for $\rm GL_2$ Maass forms, the Rankin-Selberg coefficients $(\lambda_f(n)^2)_{n\geq 1}$ admit a level of distribution $\theta=2/5+1/260-\eta$ in arithmetic progressions.
We initiate the study of certain families of $L$-functions attached to characters of subgroups of higher-rank tori, and of their average at the central point. In particular, we evaluate the average of the values $L(\demi,\chi^a)L(\demi,\chi^b)$ for arbitrary integers~$a$ and~$b$ when $\chi$ varies over Dirichlet characters to a prime modulus.
In this paper we study the subconvexity problem for the Rankin-Selberg L-function and triple product L-function, allowing joint ramifications and conductor dropping range. We first extend the method of Michel-Venkatesh to reduce the bounds for L-functions to local conjectures on test vectors, then verify these local conjectures under certain conditions, giving new subconvex bounds as long as the representations are not completely related.
We prove that the coefficients of a $\mathrm{GL}_3\times \mathrm{GL}_2$ Rankin--Selberg $L$-function do not correlate with a wide class of trace functions of small conductor modulo primes, generalizing the corresponding result \cite{FKM1} for~$\mathrm{GL}_2$ and \cite{KLMS} for $\mathrm{GL}_3$. This result is inspired by a recent work of P. Sharma who discussed the case of a Dirichlet character of prime modulus.
We study the simultaneous reductions at several supersingular primes of elliptic curves with complex multiplication. We show -- under additional congruence assumptions on the CM order -- that the reductions are surjective (and even become equidistributed) on the product of supersingular loci when the discriminant of the order becomes large. This variant of the equidistribution theorems of Duke and Cornut-Vatsal is an(other) application of the recent work of Einsiedler and Lindenstrauss on the classification of joinings of higher-rank diagonalizable actions.
In the analytic study of trace functions of $\ell$-adic sheaves over finite fields, a crucial issue is to control the conductor of sheaves constructed in various ways. We consider cohomological transforms on the affine line over a finite field which have trace functions given by linear operators with an additive character of a rational function in two variables as a kernel. We prove that the conductor of such a transform is bounded in terms of the complexity of the input sheaf and of the rational function defining the kernel, and discuss applications of this result, including motivating examples arising from the Polymath8 project.
We prove that sums of length about $q^{3/2}$ of Hecke eigenvalues of automorphic forms on $SL_3(\Zz)$ do not correlate with $q$-periodic functions with bounded Fourier transform. This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.
For a fairly general family of L-functions, we survey the known consequences of the existence of asymptotic formulas with power-sawing error term for the (twisted) first and second moments of the central values in the family. We then consider in detail the important special case of the family of twists of a fixed cusp form by primitive Dirichlet characters modulo a prime q, and prove that it satisfies such formulas. We derive arithmetic consequences: - a positive proportion of central values L(f x chi, 1/2) are non-zero, and indeed bounded from below; - there exist many characters chi for which the central L-value is very large; - the probability of a large analytic rank decays exponentially fast. We finally show how the second moment estimate establishes a special case of a conjecture of Mazur and Rubin concerning the distribution of modular symbols.
We describe how a systematic use the deep methods from $\ell$-adic cohomology pioneered by Grothendieck and Deligne and further developed by Katz, Laumon allow to make progress on various classical questions from analytic number theory. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 Arizona Winter School.
We revisit a recent bound of I. Shparlinski and T. P. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet $L$-functions.