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

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

A note on the Mordell-Weil rank modulo n

Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is conjecturally the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.Comment: 7 page

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

LLL & ABC

This note is an observation that the LLL algorithm applied to prime powers can be used to find "good" examples for the ABC and Szpiro conjectures.Comment: 6 pages; record algebraic example included; final version, to appear in J. Number Theor