2,611 research outputs found

### Endomorphisms of abelian varieties, cyclotomic extensions and Lie algebras

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

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

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

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

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

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

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 $K$-theory of Discrete Groups

Let $\Gamma$ 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 $\Gamma$, determined on its elements of
finite order, which is of finite type. Then we determine the contribution of
this ring to the topological $K$-theory $K^*(B\Gamma)$, obtaining an exact
formula for the difference in terms of the cohomology of the centralizers of
elements of finite order in $\Gamma$.Comment: 4 page

### Serre's "formule de masse" in prime degree

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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