Let f be a Hecke eigenform of even integral weight on the full modular group, and let λ _sym^jf(n) denote the coefficients of the j-th symmetric power L-function attached to f. Extending recent results of Sharma and Sankaranarayanan (2022–2023), which treat the first and second power moments of these coefficients, we study the higher power moments corresponding to the third and fourth powers for j≥ 2. We obtain asymptotic estimates with refined error terms and establish generalizations over suitable arithmetic sets.
The classes of abelian groups that are (uniformly) strongly Hopfian abelian groups and, dually, (uniformly) strongly co-Hopfian abelian groups have been intensively studied by several authors, including Abdelalim–Chillali–Essanouni [1] and Abdelalim [2]. This paper extends these investigations in the case of (genuinely) mixed groups. For example, it is shown that a reduced group that is (uniformly) strongly co-Hopfian will always be (uniformly) strongly Hopfian. In addition, a result of Chekhlov–Danchev [4] characterizing when a torsion-free group is uniformly strongly Hopfian is generalized to the case of (global) Warfield groups.
Let f be an isometry of a median metric space. We show that if f attains its minimal displacement and no power of f has an inversion then f admits an axis. This generalizes to the median setting a result on isometries of CAT(0) cube complexes.
We develop a framework that enables us to study a broad class of special values of the Katz two-variable p-adic L-function, including certain special values lying outside the range of p-adic interpolation.
We present two geometrizations for the group of p-adic characters of the additive group of the ring ℤ_p of p-adic integers. The first geometrization is realized via certain étale sheaves on the Witt ring scheme. The second is constructed in terms of certain quasi-coherent sheaves, equipped with an additional structure, on the formal completion of the Witt ring scheme along its special fiber.
Inspired by a beautiful formula of Bertolini, Darmon, and Prasanna—the oft-termed BDP formula—we address questions about the non-vanishing of non-torsion points under p-adic logarithms of abelian varieties. We largely consider situations most applicable to GL_2 -type abelian varieties associated with Hilbert modular newforms and Heegner points. Not surprisingly, the main tool employed is the p-adic analytic subgroup theorem.
In this note we show that certain meromorphic orthogonal modular forms are magnetic, i.e. their Fourier coefficients satisfy special divisibility criteria. These meromorphic orthogonal modular forms are counterparts to the orthogonal cusp forms considered by Oda. We show that the seminal work of Borcherds implies the magneticity of these forms.
We establish two new universal inequalities for Neumann eigenvalues of the Laplacian on a planar convex domain.
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology. On the other hand, there are several variants of these notions: with open covers or else Whitehead–Ganea constructions, also with spaces and maps that are unpointed or else pointed, fibrewise, equivariant, etc. or even with algebraic models of spaces and maps. These are two reasons why we build an axiomatic approach to all these notions, based on just three simple axioms. We also introduce the notion of ‘lifting category’ which unifies the notions of sectional category, topological complexity, and homotopic distance, all of which are special cases of lifting category.
We study the Doubrov–Zelenko symplectification procedure for rank 2 distributions with 5-dimensional cube—originally motivated by optimal control theory—through the lens of Tanaka–Morimoto theory for normal Cartan connections. In this way, for ambient manifolds of dimension n ≥ 5 , we prove the existence of the normal Cartan connection associated with the symplectified distribution. Furthermore, we show that this symplectification can be interpreted as the (n-4) th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank 2 distribution with 5-dimensional cube: (1) Is the (n-4) th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the (n-4) th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka–Morimoto theory? Our main results demonstrate that: (a) For n > 5 , the answer to the second question is positive (in contrast to the classical n = 5 case from G_2 -parabolic geometries); (b) For n ≥ 5 , the answer to the first question is negative: unification occurs already at the (n-5) th iterated Cartan prolongation.
We introduce an axiomatization of the notion of ( p-complete) anticyclotomic Euler system for a wide class of Galois representations, including those attached to a cuspidal eigenform and to a Hida family of modular forms. Under a minimal set of assumptions, we show how to build from these data a universal Kolyvagin system for the representation and for its anticyclotomic twist. Eventually, we recover some applications to the structure of Selmer groups and Iwasawa main conjectures and we review a few concrete examples of these abstract notions that can be found in the literature.
The notion of double depth associated with quasi-Jacobi forms allows distinguishing, within the algebra JS^∞ of quasi-Jacobi singular forms of index zero, certain significant subalgebras (modular-type forms, elliptic-type forms, Jacobi forms). We study the stability of these subalgebras under the derivations of JS^∞ and through certain sequences of bidifferential operators constituting analogs of Rankin–Cohen brackets or transvectants.
This article provides partial solutions to Chinburg’s conjectures by studying a sequence of multivariate polynomials. These conjectures assert that for every odd quadratic Dirichlet character of conductor f, χ _-f=( -f/.) , there exists a bivariate polynomial (or a rational function in the weak version) whose Mahler measure is a rational multiple of L'(χ _-f,-1) . We prove that the Mahler measure of a polynomial family, denoted by P_d , can be expressed as a linear combination of the derivatives of Dirichlet L-functions. Specifically, this family provides solutions to the conjectures for conductors f=3,4,8,15,20 , and 24. We further generalize Chinburg’s conjectures from real primitive odd Dirichlet characters to all primitive odd characters. For this generalized version, the polynomials P_d provide solutions for conductors 5, 7, and 9.
Let K = ℚ(i) . We study the Petersson inner product of a Hermitian Eisenstein series of Siegel type on the unitary group U_5(K) , diagonally-restricted on U_2(K)× U_2(K)× U_1(K) , against two Hermitian cuspidal eigenforms F, G of degree 2 and an elliptic cuspidal eigenform h (seen as a Hermitian modular form of degree 1), all having weight k ≡ 0 4 . We obtain, through this consideration, an integral representation of a certain Dirichlet series, together with an additional residue term. By taking F to belong in the Maass space, we are able to show that the Dirichlet series possesses an Euler product. Moreover, its p-factor for an inert prime p can be essentially identified with the twist by h of a degree six Euler factor attached to G by Gritsenko. The question of whether the same holds for the primes that split remains unanswered here, even though we make considerable steps in that direction too. Our paper is inspired by a work of Heim, who considered a similar question in the case of Siegel modular forms.
This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local information that is not visible when looking at the objects from a commutative angle. For instance, it is a precise meaning to be given to two things being ``infinitesimally close'', something being obscured from view when restricting only to a commutative algebraic study. A Platonesque way of looking at this is that the commutative world is a ``shadow'' of a more inclusive non-commutative universe. The present paper aims at laying the foundation for further and deeper study of arithmetic and geometry using non-commutative geometry and non-commutative deformation theory.
This paper is a follow-up on the \emph{noncommutative differential geometry on infinitesimal spaces} [15]. In the present work, we extend the algebraic convergence from [15] to the geometric setting. On the one hand, we reformulate the definition of finite dimensional compatible Dirac operators using Clifford algebras. This definition also leads to a new construction of a Laplace operator. On the other hand, after a well-chosen Green's function defined on a manifold, we show that when the Dirac operators can be interpreted as stochastic matrices. The sequence $(D_n)_{n\in \mathbb{N}}$ converges then in average to the usual Dirac operator on a spin manifold. The same conclusion can be drawn for the Laplace operator.
The purpose of this paper is to study the special values of the standard L-functions for quaternionic modular forms using the doubling method. We obtain an integral representation for the L-function twisted by a character and construct the p-adic measure interpolating certain special L-values.
Ce papier est une incursion de la Théorie des Modèles dans la Géométrie Algébrique, et son auteur attache un plus grand prix aux méthodes employées pour parvenir aux résultats obtenus qu'à la valeur intrinsèque de ces résultats-mêmes. En témoigne la nature des questions réparties dans le texte. On y étudie les influences réciproques du groupe des automorphismes d'un corps algébriquement clos K sur le groupe des automorphismes d'une structure S définissable dans K. Les paramètres nécessaires aux définitions vont y jouer un rôle de premier plan, ainsi que les propriétés très particulières de la Théorie des Modèles des corps algébriquement clos. Il commence par un commentaire d’un résultat de A.V. Borovik, qui a été la source de son inspiration, mettant en évidence sa dépendance au célèbre Théorème de Borel et Tits sur les isomorphismes abstraits des groupes algébriques simples, considéré d’un point de vue modèle-théorique. Ce théorème conduit finalement à la description des automorphismes d'ordre fini d'un groupe algébrique simple (sur un corps de base algébriquement clos), et de ses groupes superstables d’automorphismes, qui est basée sur des arguments généraux de Théorie des Modèles, ne demandant qu’une inspection minimale de la structure du groupe; pour en tirer des conséquences, nous devons affermir un argument elliptique d’Altinel, Borovik et Cherlin, dans une démonstration qui est pourtant cruciale dans leur contexte inductif. En fait, notre version du Théorème de Borel et Tits ne dépend pas de la présence d’une loi de groupe; il est valable plus généralement pour ce que nous appelons les structures constructives autonomes, qui sont les structures infinies S, définissables dans un corps algébriquement clos K, pour lesquelles tout ce qui est définissable sur S dans le langage du corps K est définissable (avec paramètres) dans le langage de S. Un exemple significatif de telles structures est donné par les multicorps, dont la banale définition cache une subtile théorie de Galois en caractéristique p; en effet, dans n’importe quelle structure constructible autonome S on peut définir sans paramètres un multicorps qui contrôle les automorphismes de S au sens suivant: ceux d’entre eux dont l’action sur ce multicorps est d’ordre fini forment un groupe, noté Autmax(S), qui est à la fois le plus grand groupe définissable d’automorphismes de S, et le plus grand groupe superstable d’automorphismes de S. En caractéristique nulle, ce résultat est facile à établir, car on peut alors définir sans paramètres dans S un unicorps, c’est-à-dire une copie L du corps de base K. Quand S est un groupe algébrique simple G, dans les cas ordinaires une telle copie L existe même en caractéristique p; mais il y a des cas spéciaux, dus à la présence d'endogénies exceptionnelles, ou seulement un bicorps (L1,L2) est définissable sans paramètres; Autmax(G) est le noyau de l’action des automorphismes de G sur le corps dans le premier cas, sur le bicorps dans le second, et il est en fait égal au groupe des automorphismes géométriques de G. L’existence de Autmax(G) s’obtient par des méthodes directes de Théorie des Modèles, mais il importe de se rendre compte que sa composante connexe est formée des automorphismes intérieurs de G. Comme il est définissable, c’est-à-dire constructiblement isomorphe à un groupe algébrique, il s’agit là d’un résultat de pure géométrie, qui est par ailleurs bien connu; sa démonstration s’appuie sur une description minutieuse de la structure de G, mais il est inutile de la reproduire pour obtenir sa généralisation modèle-théorique, à savoir que, dans un contexte de rang de Morley fini, tout groupe connexe d’automorphismes de G est formé d’automorphismes intérieurs.
Let G be a complex reductive group and D ⊂ X a finite subset of a compact Riemann surface X. It was shown in Biswas and Jeffrey (Ann Math Québec 45: 213–219, 2021) that the moduli space of G–characters of π _1(X∖ D) has a natural Poisson structure. We show that the moduli space of logarithmic G–connections on X singular over D has a Poisson structure. It is proved that the monodromy map from the moduli space of logarithmic G–connections to the moduli space of G–characters is Poisson structure preserving.
