47 research outputs found
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
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
Tate module and bad reduction
Let C/K be a curve over a local field. We study the natural semilinear action
of Galois on the minimal regular model of C over a field F where it becomes
semistable. This allows us to describe the Galois action on the l-adic Tate
module of the Jacobian of C/K in terms of the special fibre of this model over
F.Comment: 13 pages, final version, to appear in Proc. AM
Self-duality of Selmer groups
The first part of the paper gives a new proof of self-duality for Selmer
groups: if A is an abelian variety over a number field K, and F/K is a Galois
extension with Galois group G, then the Q_pG-representation naturally
associated to the p-infinity Selmer group of A/F is self-dual. The second part
describes a method for obtaining information about parities of Selmer ranks
from the local Tamagawa numbers of A in intermediate extensions of F/K.Comment: 12 pages; to appear in Proc. Cam. Phil. So