On the Fontaine-mazur Conjecture for Number Fields and an Analogue for Function Fields
نویسنده
چکیده
The Fontaine-Mazur Conjecture for number fields predicts that infinite `-adic analytic groups cannot occur as the Galois groups of unramified `-extensions of number fields. We investigate the analogous question for function fields of one variable over finite fields, and then prove some special cases of both the number field and function field questions using ideas from class field theory, `-adic analytic groups, Lie algebras, arithmetic algebraic geometry, and Iwasawa theory.
منابع مشابه
Notes on an analogue of the Fontaine - Mazur conjecture par
We estimate the proportion of function fields satisfying certain conditions which imply a function field analogue of the Fontaine-Mazur conjecture. As a byproduct, we compute the fraction of abelian varieties (or even Jacobians) over a finite field which have a rational point of order l.
متن کاملAn Analogue of the Field-of-norms Functor and of the Grothendieck Conjecture
The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved...
متن کاملAn Analogue of the Field-of-norms Functor and the Grothendieck Conjecture
The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved...
متن کاملA Quantitative Fontaine-mazur Analogue for Function Fields
Let k be a function field over a finite field F of characteristic p and order q, and l a prime not equal to p. Let K = kFl∞ be obtained from k by taking the maximal l-extension of the constant field. If M is an unramified l-adic analytic l-extension of k, and M does not contain K, must M be a finite extension of k? The answer is, in general, “no”, but for some k the answer is “yes”. We attempt ...
متن کاملProof of an Exceptional Zero Conjecture for Elliptic Curves over Function Fields
Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refine...
متن کامل