Quotients of functors of Artin rings

In infinitesimal deformation theory, a classical criterion due to Schlessinger gives an intrinsic characterisation of functors that are pro-representable, and more generally, of the ones that have a hull. Our result is that in this setting the question of characterising group quotients can also be answered. In other words, for functors of Artin rings that have a hull, those that are quotients of pro-representable ones by a constant group action can be described intrinsically.Comment: 4 page

Surjectivity of mod 2^n representations of elliptic curves

For an elliptic curve E over Q, the Galois action on the l-power torsion points defines representations whose images are subgroups of GL_2(Z/l^n Z). There are three exceptional prime powers l^n=2,3,4 when surjectivity of the mod l^n representation does not imply that for l^(n+1). Elliptic curves with surjective mod 3 but not mod 9 representation have been classified by Elkies. The purpose of this note is to do this in the other two cases.Comment: 3 page

Local invariants of isogenous elliptic curves

We investigate how various invariants of elliptic curves, such as the discriminant, Kodaira type, Tamagawa number and real and complex periods, change under an isogeny of prime degree p. For elliptic curves over l-adic fields, the classification is almost complete (the exception is wild potentially supersingular reduction when l=p), and is summarised in a table.Comment: 22 pages, final version, to appear in Trans. Amer. Math. So

Solomon's induction in quasi-elementary groups

Given a finite group G, we address the following question: which multiples of the trivial representation are linear combinations of inductions of trivial representations from proper subgroups of G? By Solomon's induction theorem, all multiples are if G is not quasi-elementary. We complement this by showing that all multiples of p are if G is p-quasi-elementary and not cyclic, and that this is best possible.Comment: 2 pages, to appear in J. Group Theor

Notes on the Parity Conjecture

This is an expository article, based on a lecture course given at CRM Barcelona in December 2009. The purpose of these notes is to prove, in a reasonably self-contained way, that finiteness of the Tate-Shafarevich group implies the parity conjecture for elliptic curves over number fields. Along the way, we review local and global root numbers of elliptic curves and their classification, and discuss some peculiar consequences of the parity conjecture.Comment: minor corrections, to appear in a CRM Advanced Courses volume "Elliptic curves, Hilbert modular forms and Galois deformations"; 43 page

On the Birch-Swinnerton-Dyer quotients modulo squares

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer rank and the analytic rank agree.Comment: 29 pages; minor changes; to appear in Annals of Mathematic

A remark on Tate's algorithm and Kodaira types

We remark that Tate's algorithm to determine the minimal model of an elliptic curve can be stated in a way that characterises Kodaira types from the minimum of v(a_i)/i. As an application, we deduce the behaviour of Kodaira types in tame extensions of local fields.Comment: 6 pages (minor changes