109 research outputs found
Connectedness of Hecke algebras and the Rayuela conjecture: a path to functoriality and modularity
No full textLet ρ1 and ρ2 be a pair of residual, odd, absolutely irreducible two-dimensional Galois representations of a totally real number field F. In this article we propose a conjecture asserting existence of "safe" chains of compatible systems of Galois representations linking ρ1 to ρ2. Such conjecture implies the generalized Serre's conjecture and is equivalent to Serre's conjecture under a modular version of it. We prove a weak version of the modular variant using the connectedness of certain Hecke algebras, and we comment on possible applications of these results to establish some cases of Langlands functoriality.Fil: Dieulefait, Luis. Universidad de Barcelona; EspañaFil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
On the change of root numbers under twisting and applications
No full textThe purpose of this article is to show how the root number of a modular form changes by twisting in terms of the local Weil-Deligne representation at each prime ideal. As an application, we show how one can for each odd prime p, determine whether a modular form (or a Hilbert modular form) with trivial nebentypus is Steinberg, Principal Series or Supercuspidal at p by analyzing the change of sign under a suitable twist. We also explain the case p = 2, where twisting is not enough in general.Fil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin
On the embedding problem for 2+s4 representations
No full textLet 2+S4 denote the double cover of S4 corresponding to the element in H2(S4, Z/2Z) where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4. Given an elliptic curve E, let E[2] denote its 2-torsion points. Under some conditions on E elements in H1(GalQ, E[2])\{0} correspond to Galois extensions N of Q with Galois group (isomorphic to) S4. In this work we give an interpretation of the addition law on such fields, and prove that the obstruction for N having a Galois extension N˜ with Gal(N/˜ Q) 2+S4 gives a homomorphism s+ 4 : H1(GalQ, E[2]) → H2(GalQ, Z/2Z). As a corollary we can prove (if E has conductor divisible by few primes and high rank) the existence of 2-dimensional representations of the absolute Galois group of Q attached to E and use them in some examples to construct 3/2 modular forms mapping via the Shimura map to (the modular form of weight 2 attached to) E.Fil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
Modularity of the Consani-Scholten Quintic
No full textWe prove that the Consani-Scholten quintic, a Calabi-Yau threefold over Q, is Hilbert modular. For this, we refine several techniques known from the context of modular forms. Most notably, we extend the Faltings-Serre-Livn ́ e method to induced four-dimensional Galois representations over Q. We also need a Sturm bound for Hilbert modular forms; this is developed in an appendix by Jose Burgos Gil and the second author.Fil: Dieulefait, Luis. No especifíca;Fil: Schutt, Matthias. No especifíca;Fil: Pacetti, Ariel Martín. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentin
Computing ideal classes representatives in quaternion algebras
No full textLet K be a totally real number field and let B be a totally definite quaternion algebra over K. Given a set of representatives for ideal classes for a maximal order in B, we show how to construct in an efficient way a set of representatives of ideal classes for any Bass order in B. The algorithm does not require any knowledge of class numbers, and improves the equivalence checking process by using a simple calculation with global units. As an application, we compute ideal classes representatives for an order of discriminant 30 in an algebra over the real quadratic field Q ([√ 5].Fil: Pacetti, Ariel Martín. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Sirolli, Nicolás Martín. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
On computing finite index subgroups of PSL 2 (ℤ)
No full textWe present a method to compute finite index subgroups of PSL2(ℤ). Our strategy follows Kulkarni’s ideas, the main contribution being a recursive method to compute bivalent trees as well as their automorphism groups. As a concrete application, we compute all subgroups of index up to 20. We then use this database to produce tables with several arithmetical properties.Fil: Mayorga Uruburu, Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; ArgentinaFil: Pacetti, Ariel Martín. Universidade de Aveiro; Portugal. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Vendramin, Claudio Leandro. Vrije Unviversiteit Brussel; Bélgica. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
Anticyclotomic p-adic L-functions for elliptic curves at some additive reduction primes
No full textLet E be a rational elliptic curve and let p be an odd prime of additive reduction. Let K be an imaginary quadratic field and fix a positive integer c prime to the conductor of E. The main goal of the present article is to define an anti-cyclotomic p-adic L-function L attached to E/K when E/Q p attains semistable reduction over an abelian extension. We prove that L satisfies the expected interpolation properties; namely, we show that if χ is an anticyclotomic character of conductor cp^n then χ(L ) is equal (up to explicit constants) to L(E, χ, 1) or L´(E, χ, 1).Soit E une courbe elliptique rationnelle et p un premier impair de réduction additive. Soit K un corps quadratique imaginaire et c un entier positif, premier au conducteur de E. Le but de cette Note est de définir une fonction L p-adique, anti-cyclotomique, notée , attachée à lorsque atteint la réduction semi-stable sur une extension abélienne. Nous montrons que satisfait les propriétés d'interpolation escomptées. Précisément, nous montrons que, si χ est un caractère anti-cyclotomique de conducteur , alors est égal (à des constantes explicites près) à ou .Fil: Kohen, Daniel. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Pacetti, Ariel Martín. Universidad Nacional de Córdoba; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
Proving Modularity for a given elliptic curve over an imaginary quadratic field
No full textWe present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor and Berger-Harcos (cf. [HST93], [Tay94] and [BH07]) we can associate to an automorphic representation a family of compatible ℓ-adic representations. Our algorithm is based on Faltings-Serre’s method to prove that ℓ-adic Galois representations are isomorphic. Using the algorithm we provide the first examples of modular elliptic curves over imaginary quadratic fields with residual 2-adic image isomorphic to S3 and C3.Fil: Dieulefait, Luis. Universidad de Barcelona; EspañaFil: Guerberoff, Lucio. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universite Paris Diderot - Paris 7; FranciaFil: Pacetti, Ariel Martín. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
Congruences between modular forms modulo prime powers
No full textGiven a prime p≥5 and an abstract odd representation ρ n with coefficients modulo p n (for some n≥1 and big image, we prove the existence of a lift of ρ n to characteristic 0 whenever local lifts exist (under minor technical conditions). Moreover, our results allow to chose the lift's inertial type at all primes but finitely many (where the lift is of Steinberg type). We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. An easy application of our main result proves that given a modular eigenform f whose Galois representation is not induced from a character (i.e., f has no inner twists), for all primes p but finitely many, and for all positive integers n, there exists an eigenform g ≠f which is congruent to f modulo pn.Fil: Camporino, Maximiliano Javier. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Pacetti, Ariel Martín. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomia y Física. Sección Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentin
Hecke and sturm bounds for Hilbert modular forms over real quadratic fields
No full textLet K be a real quadratic field and OK its ring of integers. Let Γ be a congruence subgroup of SL2(OK) and M(k1,k2)(Γ) be the finite dimensional space of Hilbert modular forms of weight (k1, k2) for Γ. Given a form f(z) ∈ M(k1,k2)(Γ), how many Fourier coefficients determine it uniquely in such space? This problem was solved by Hecke for classical forms, and Sturm proved its analogue for congruences modulo a prime ideal. The present article solves the same problem for Hilbert modular forms over K. We construct a finite set of indices (which depends on the cusps desingularization of the modular surface attached to Γ) such that the Fourier coefficients of any form in such set determines it uniquely.Fil: Burgos Gil, Jose Ignacio. Instituto de Ciencias Matemáticas; EspañaFil: Pacetti, Ariel Martín. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin
- …
