2,611 research outputs found

    Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

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    We prove an analogue of the Tate conjecture on homomorphisms of abelian varieties over infinite cyclotomic extensions of finitely generated fields of characteristic zero.Comment: 9 page

    Serre Theorem for involutory Hopf algebras

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    We call a monoidal category C{\mathcal C} a Serre category if for any CC, D∈CD \in {\mathcal C} such that C\ot D is semisimple, CC and DD are semisimple objects in C{\mathcal C}. Let HH be an involutory Hopf algebra, MM, NN two HH-(co)modules such that M⊗NM \otimes N is (co)semisimple as a HH-(co)module. If NN (resp. MM) is a finitely generated projective kk-module with invertible Hattory-Stallings rank in kk then MM (resp. NN) is (co)semisimple as a HH-(co)module. In particular, the full subcategory of all finite dimensional modules, comodules or Yetter-Drinfel'd modules over HH the dimension of which is invertible in kk are Serre categories.Comment: a new version: 8 page

    Automorphisms and forms of simple infinite-dimensional linearly compact Lie superalgebras

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    We describe the group of continuous automorphisms of all simple infinite-dimensional linearly compact Lie superalgebras and use it in order to classify F-forms of these superalgebras over any field F of characteristic zero.Comment: 24 page

    Some genus 3 curves with many points

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    Using an explicit family of plane quartic curves, we prove the existence of a genus 3 curve over any finite field of characteristic 3 whose number of rational points stays within a fixed distance from the Hasse-Weil-Serre upper bound. We also provide an intrinsic characterization of so-called Legendre elliptic curves

    Non-commutative p-adic L-functions for supersingular primes

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    Let E/Q be an elliptic curve with good supersingular reduction at p with a_p(E)=0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the Zp-cyclotomic extension of a finite Galois extension of Q where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.Comment: 13 pages; some minor corrections; to appear in International Journal of Number Theor

    Diversity in Parametric Families of Number Fields

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    Let X be a projective curve defined over Q and t a non-constant Q-rational function on X of degree at least 2. For every integer n pick a point P_n on X such that t(P_n)=n. A result of Dvornicich and Zannier implies that, for large N, among the number fields Q(P_1),...,Q(P_N) there are at least cN/\log N distinct, where c>0. We prove that there are at least N/(\log N)^{1-c} distinct fields, where c>0.Comment: Minor inaccuracies detected by the referees are correcte

    Representations and KK-theory of Discrete Groups

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    Let Γ\Gamma be a discrete group of finite virtual cohomological dimension with certain finiteness conditions of the type satisfied by arithmetic groups. We define a representation ring for Γ\Gamma, determined on its elements of finite order, which is of finite type. Then we determine the contribution of this ring to the topological KK-theory K∗(BΓ)K^*(B\Gamma), obtaining an exact formula for the difference in terms of the cohomology of the centralizers of elements of finite order in Γ\Gamma.Comment: 4 page

    Serre's "formule de masse" in prime degree

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    For a local field F with finite residue field of characteristic p, we describe completely the structure of the filtered F_p[G]-module K^*/K^*p in characteristic 0 and $K^+/\wp(K^+) in characteristic p, where K=F(\root{p-1}\of F^*) and G=\Gal(K|F). As an application, we give an elementary proof of Serre's mass formula in degree p. We also determine the compositum C of all degree p separable extensions with solvable galoisian closure over an arbitrary base field, and show that C is K(\root p\of K^*) or K(\wp^{-1}(K)) respectively, in the case of the local field F. Our method allows us to compute the contribution of each character G\to\F_p^* to the degree p mass formula, and, for any given group \Gamma, the contribution of those degree p separable extensions of F whose galoisian closure has group \Gamma.Comment: 36 pages; most of the new material has been moved to the new Section
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