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Barry Mazur

Publications and source records attributed to Barry Mazur.

24 records · Page 2Linked to original sources
arXiv2008

Nearly Ordinary Galois Deformations over Arbitrary Number Fields

Let K be an arbitrary number field, and let rho: Gal(Kbar/K) -> GL_2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of rho. When K is totally real and rho is modular, results of Hida imply that the nearly ordinary deformation space associated to rho contains a Zariski dense set of points corresponding to "automorphic" Galois representations. We conjecture that if K is_not_ totally real, then this is never the case, except in three exceptional cases, corresponding to (1) "base change", (2) "CM" forms, and (3) "Even" representations. The latter case conjecturally can only occur if the image of rho is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of Leopoldt's conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of "many" positive dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about "p-adic functorality", as well as some remarks on how our methods should apply to n-dimensional representations of Gal(Qbar/Q) when n > 2.

math.NT↗
arXiv2007

Growth of Selmer rank in nonabelian extensions of number fields

Let p be an odd prime number, E an elliptic curve over a number field k, and F/k a Galois extension of degree twice a power of p. We study the Z_p-corank rk_p(E/F) of the p-power Selmer group of E over F. We obtain lower bounds for rk_p(E/F), generalizing the results in [MR], which applied to dihedral extensions. If K is the (unique) quadratic extension of k in F, G = Gal(F/K), G^+ is the subgroup of elements of G commuting with a choice of involution of F over k, and rk_p(E/K) is odd, then we show that (under mild hypotheses) rk_p(E/F) \ge [G:G^+]$. As a very specific example of this, suppose A is an elliptic curve over Q with a rational torsion point of order p, and with no complex multiplication. If E is an elliptic curve over Q with good ordinary reduction at p, such that every prime where both E and A have bad reduction has odd order in F_p^\times, and such that the negative of the conductor of E is not a square mod p, then there is a positive constant B, depending on A but not on E or n, such that rk_p(E/Q(A[p^n])) \ge B p^{2n} for every n.

math.NT↗
arXiv2006

Finding large Selmer rank via an arithmetic theory of local constants

We obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields. Suppose $K/k$ is a quadratic extension of number fields, $E$ is an elliptic curve defined over $k$, and $p$ is an odd prime. Let $F$ denote the maximal abelian $p$-extension of $K$ that is unramified at all primes where $E$ has bad reduction and that is Galois over $k$ with dihedral Galois group (i.e., the generator $c$ of $Gal(K/k)$ acts on $Gal(F/K)$ by -1). We prove (under mild hypotheses on $p$) that if the rank of the pro-$p$ Selmer group $S_p(E/K)$ is odd, then the rank of $S_p(E/L)$ is at least $[L:K]$ for every finite extension $L$ of $K$ in $F$.

math.NT↗
arXiv2003

Elliptic curves and class field theory

Suppose $E$ is an elliptic curve defined over $\Q$. At the 1983 ICM the first author formulated some conjectures that propose a close relationship between the explicit class field theory construction of certain abelian extensions of imaginary quadratic fields and an explicit construction that (conjecturally) produces almost all of the rational points on $E$ over those fields. Those conjectures are to a large extent settled by recent work of Vatsal and of Cornut, building on work of Kolyvagin and others. In this paper we describe a collection of interrelated conjectures still open regarding the variation of Mordell-Weil groups of $E$ over abelian extensions of imaginary quadratic fields, and suggest a possible algebraic framework to organize them.

math.NT↗
arXiv1995

Evolutions, Symbolic Squares, and Fitting Ideals

Given a reduced local algebra $T$ over a suitable ring or field $k$ we study the question of whether there are nontrivial algebra surjections $R\to T$ which induce isomorphisms $Ω_{R/k}\otimes T \to Ω_{T/k}$. Such maps, called evolutions, arise naturally in the study of Hecke algebras, as they implicitly do in the recent work of Wiles, Taylor-Wiles, and Flach. We show that the existence of non-trivial evolutions of an algebra $T$ can be characterized in terms of the symbolic square of an ideal defining $T$. We give a characterization of the symbolic square in terms of Fitting ideals. Using this and other techniques we show that certain classes of reduced algebras -- codimension 2 Cohen-Macaulay, Codimension 3 Gorenstein, licci algebras in general, and some others -- admit no nontrivial evolutions. On the other hand we give examples showing that non-trivial evolutions of reduced Cohen-Macaulay algebras of codimension 3 do exist in every positive characteristic.

alg-geom↗