50 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
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
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
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