المؤلفون: Blomer, Valentin, Harcos, Gergely, Maga, Péter, Milićević, Djordje

المصدر: J. Math. Pures Appl. 168 (2022), 1-64

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - Spectral Theory

الوصف: We open a new perspective on the sup-norm problem and propose a version for non-spherical Maass forms when the maximal compact K is non-abelian and the dimension of the K-type gets large. We solve this problem for an arithmetic quotient of G=SL_2(C) with K=SU_2(C). Our results cover the case of vector-valued Maass forms as well as all the individual scalar-valued Maass forms of the Wigner basis, reaching sub-Weyl exponents in some cases. On the way, we develop analytic theory of independent interest, including uniform strong localization estimates for generalized spherical functions of high K-type and a Paley-Wiener theorem for the corresponding spherical transform acting on the space of rapidly decreasing functions. The new analytic properties of the generalized spherical functions lead to novel counting problems of matrices close to various manifolds that we solve optimally.
Comment: 68 pages, LaTeX2e; to appear in Journal de Math\'ematiques Pures et Appliqu\'ees

الوصول الحر: http://arxiv.org/abs/2107.05973Test

المؤلفون: Brumley, Farrell, Lesesvre, Didier, Milićević, Djordje

مصطلحات موضوعية: Mathematics - Number Theory, 11F41, 11F72

الوصف: We obtain a Weyl law with power savings for the universal families of cuspidal automorphic representations, ordered by analytic conductor, of $\mathrm{GL}_2$ over $\mathbb{Q}$, as well as for Hecke characters over any number field. The method proceeds by establishing the requisite analytic properties of the underlying conductor zeta function.
Comment: 26 pages

الوصول الحر: http://arxiv.org/abs/2105.02068Test

المؤلفون: Frączyk, Mikołaj, Harcos, Gergely, Maga, Péter, Milićević, Djordje

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - Spectral Theory, Primary 11F72, Secondary 11F70, 22E40, 22E45

الوصف: We prove the density hypothesis for wide families of arithmetic orbifolds arising from all division quaternion algebras over all number fields of bounded degree. Our power-saving bounds on the multiplicities of non-tempered representations are uniform in the volume and spectral aspects.
Comment: 34 pages, LaTeX2e

الوصول الحر: http://arxiv.org/abs/2007.13961Test

المؤلفون: Milićević, Djordje, Zhang, Sichen

مصطلحات موضوعية: Mathematics - Number Theory, 11L05 (Primary) 11T23, 60F17, 60G16, 60G50 (Secondary)

الوصف: We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p^n of a fixed odd prime p, a pure depth-aspect analogue of theorems of Kowalski-Sawin and Ricotta-Royer-Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as n->\infty to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every p converges in law as p->\infty, and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the p-adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments.
Comment: 35 pages

الوصول الحر: http://arxiv.org/abs/2005.08865Test

المؤلفون: Dever, Lindsay, Milićević, Djordje

المصدر: International Mathematics Research Notices (2021), rnab048

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - Differential Geometry, 11F72 (Primary) 53C22, 53C29, 58J50 (Secondary)

الوصف: We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy and have both the rate of shrinking and the strength of the error term fully symmetric in length and holonomy.
Comment: 39 pages, final version, IMRN

الوصول الحر: http://arxiv.org/abs/2004.11774Test

المؤلفون: Khan, Rizwanur, Milićević, Djordje, Ngo, Hieu T.

مصطلحات موضوعية: Mathematics - Number Theory

الوصف: We show that for at least $\frac{5}{13}$ of the primitive Dirichlet characters $\chi$ of large prime modulus, the central value $L(\frac{1}{2},\chi)$ does not vanish, improving on the previous best known result of $\frac{3}{8}$.
Comment: Improved proportion of nonvanishing. Added an author

الوصول الحر: http://arxiv.org/abs/1911.10268Test

المؤلفون: Milićević, Djordje, White, Daniel

مصطلحات موضوعية: Mathematics - Number Theory, 11M06, 11L07 (Primary), 11L40, 26E30 (Secondary)

الوصف: We prove the q-aspect analogue of Heath-Brown's result on the twelfth power moment of the Riemann zeta function for Dirichlet L-functions to odd prime power moduli. Our results rely on the p-adic method of stationary phase for sums of products and complement Nunes' bound for smooth square-free moduli.
Comment: minor revision incorporating referee's comments; final version to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze

الوصول الحر: http://arxiv.org/abs/1908.04833Test

المؤلفون: Brumley, Farrell, Milićević, Djordje

مصطلحات موضوعية: Mathematics - Number Theory, Mathematics - Representation Theory, 11F72, 11F66, 11F70, 22E55, 58J50

الوصف: Let $F$ be a number field and $n\geqslant 1$ an integer. The universal family is the set $\mathfrak{F}$ of all unitary cuspidal automorphic representations on ${\rm GL}_n$ over $F$, ordered by their analytic conductor. We prove an asymptotic for the size of the truncated universal family $\mathfrak{F}(Q)$ as $Q\rightarrow\infty$, under a spherical assumption at the archimedean places when $n\geqslant 3$. We interpret the leading term constant geometrically and conjecturally determine the underlying Sato--Tate measure. Our methods naturally provide uniform Weyl laws with logarithmic savings in the level and strong quantitative bounds on the non-tempered discrete spectrum for ${\rm GL}_n$.
Comment: 103 pages, 5 figures, to appear in Annales Scientifiques de l'\'Ecole Normale Sup\'erieure

الوصول الحر: http://arxiv.org/abs/1805.00633Test

المؤلفون: Blomer, Valentin, Fouvry, Étienne, Kowalski, Emmanuel, Michel, Philippe, Milićević, Djordje, Sawin, Will

مصطلحات موضوعية: Mathematics - Number Theory

الوصف: 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.
Comment: 154 pages, small updates relative to the first version, to appear in Memoirs AMS

الوصول الحر: http://arxiv.org/abs/1804.01450Test

المؤلفون: Blomer, Valentin, Harcos, Gergely, Maga, Péter, Milićević, Djordje

المصدر: J. Eur. Math. Soc. (JEMS) 22 (2020) 1-53

مصطلحات موضوعية: Mathematics - Number Theory, 11F72, 11F55, 11J25

الوصف: We solve the sup-norm problem for spherical Hecke-Maass newforms of square-free level for the group GL(2) over a number field, with a power saving over the local geometric bound simultaneously in the eigenvalue and the level aspect. Our bounds feature a Weyl-type exponent in the level aspect, they reproduce or improve upon all known special cases, and over totally real fields they are as strong as the best known hybrid result over the rationals.
Comment: 40 pages, LaTeX2e; v2: revised version incorporating suggestions by the referee, to appear in JEMS

الوصول الحر: http://arxiv.org/abs/1605.09360Test