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Register a new userAbelian varieties over number fields, tame ramification and big Galois image
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Given a natural number n and a number field K, we show the existence of an integer ell_0 such that for any prime number ellgeq ell_0, there exists a finite extension F/K, unramified in all places above ell, together with a principally polarized abelian variety A of dimension n over F such that the resulting ell-torsion representation rho_{A,ell} from G_F to GSp(A[ell](bar{F})) is surjective and everywhere tamely ramified. In particular, we realize GSp_{2n}(mathbb{F}_ell) as the Galois group of a finite tame extension of number fields F/F such that F is unramified above ell.
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- Let p be a prime number and K an algebraic number field. What is the arithmetic structure of Galois extensions L/K having p-adic analytic Galois group $Gamma$ = Gal(L/K)? The celebrated Tame Fontaine-Mazur conjecture predicts that such extensions are either deeply ramified (at some prime dividing p) or ramified at an infinite number of primes. In this work, we take up a study (initiated by Boston) of this type of question under the assumption that L is Galois over some subfield k of K such that [K : k] is a prime = p. Letting $sigma$ be a generator of Gal(K/k), we study the constraints posed on the arithmetic of L/K by the cyclic action of $sigma$ on $Gamma$, focusing on the critical role played by the fixed points of this action, and their relation to the ramification in L/K. The method of Boston works only when there are no non-trivial fixed points for this action. We show that even in the presence of arbitrarily many fixed points, the action of $sigma$ places severe arithmetic conditions on the existence of finitely and tamely ramified uniform p-adic analytic extensions over K, which in some instances leads us to be able to deduce the non-existence of such extensions over K from their non-existence over k.
Let $K$ be a number field and $S$ a finite set of places of $K$. We study the kernels $Sha_S$ of maps $H^2(G_S,fq_p) rightarrow oplus_{vin S} H^2(G_v,fq_p)$. There is a natural injection $Sha_S hookrightarrow CyB_S$, into the dual $CyB_S$ of a certain readily computable Kummer group $V_S$, which is always an isomorphism in the wild case. The tame case is much more mysterious. Our main result is that given a finite $X$ coprime to $p$, there exists a finite set of places $S$ coprime to $p$ such that $Sha_{Scup X} stackrel{simeq}{hookrightarrow} CyB_{Scup X} stackrel{simeq}{twoheadleftarrow} CyB_X hookleftarrow Sha_X$. In particular, we show that in the tame case $Sha_Y$ can {it increase} with increasing $Y$. This is in contrast with the wild case where $Sha_Y$ is nonincreasing in size with increasing $Y$.
Let $[X,lambda]$ be a principally polarized abelian variety over a finite field with commutative endomorphism ring; further suppose that either $X$ is ordinary or the field is prime. Motivated by an equidistribution heuristic, we introduce a factor $ u_v([X,lambda])$ for each place $v$ of $mathbb Q$, and show that the product of these factors essentially computes the size of the isogeny class of $[X,lambda]$. The derivation of this mass formula depends on a formula of Kottwitz and on analysis of measures on the group of symplectic similitudes, and in particular does not rely on a calculation of class numbers.
Let E/F be a CM field split above a finite place v of F, let l be a rational prime number which is prime to v, and let S be the set of places of E dividing lv. If E_S denotes a maximal algebraic extension of E unramified outside S, and if u is a place of E dividing v, we show that any field embedding E_S to bar{E_u} has a dense image. The unramified outside S number fields we use are cut out from the l-adic cohomology of the simple Shimura varieties studied by Kottwitz and Harris-Taylor. The main ingredients of the proof are then the local Langlands correspondence for GL_n, the main global theorem of Harris-Taylor, and the construction of automorphic representations with prescribed local behaviours. We explain how stronger results would follow from the knowledge of some expected properties of Siegel modular forms, and we discuss the case of the Galois group of a maximal algebraic extension of Q unramified outside a single prime p and infinity.
Let $C$ be a smooth projective curve defined over a number field $k$, $X/k(C)$ a smooth projective curve of positive genus, $J_X$ the Jacobian variety of $X$ and $(tau,B)$ the $k(C)/k$-trace of $J_X$. We estimate how the rank of $J_X(k(C))/tau B(k)$ varies when we take an unramified abelian cover $pi:Cto C$ defined over $k$.
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