13 research outputs found
Wild Galois Representations: Elliptic curves over a -adic field
Given an elliptic curve over a local field with residue
characteristic , we investigate the action of the absolute Galois group of
in the case of potentially good reduction. In particular the only not
completely known case is that of the -adic Galois representation attached
to an elliptic curve such that the image of inertia is non-cyclic, and
isomorphic to . In this work we describe such a representation
explicitly.Comment: 16 pages, final version to appear in Acta Arithmetic
Wild Galois representations: elliptic curves over a -adic field with non-abelian inertia action
In this paper we present a description of the Galois representation attached
to an elliptic curve defined over a -adic field , in the case where the
image of inertia is non-abelian. There are two possibilities for the image of
inertia, namely and , and in each case we need to
distinguish whether the inertia degree of over is even or
odd. The result presented here can be implemented in an algorithm to compute
explicitly the Galois representation in these four cases.Comment: Final accepted version, to appear in IJNT. No diagrams in this
version. 10 page
Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves
Diophantine equations are a popular and active area of research in number
theory. In this paper we consider Mordell equations, which are of the form
, where is a (given) nonzero integer number and all solutions in
integers and have to be determined. One non-elementary approach for
this problem is the resolution via descent and class groups. Along these lines
we formalized in Lean 3 the resolution of Mordell equations for several
instances of . In order to achieve this, we needed to formalize several
other theories from number theory that are interesting on their own as well,
such as ideal norms, quadratic fields and rings, and explicit computations of
the class number. Moreover we introduced new computational tactics in order to
carry out efficiently computations in quadratic rings and beyond.Comment: 14 pages. Submitted to CPP '23. Source code available at
https://github.com/lean-forward/class-group-and-mordell-equatio
Power values of power sums: a survey
Research on power values of power sums has gained much attention of late,
partially due to the explosion of refinements in multiple advanced tools in
(computational) Number Theory in recent years. In this survey, we present the
key tools and techniques employed thus far in the (explicit) resolution of
Diophantine problems, as well as an overview of existing results. We also state
some open problems that naturally arise in the process.Comment: This collaboration was formed from the Women in Numbers Europe 4
worksho
Reduction Types of Genus-3 Curves in a Special Stratum of their Moduli Space
We study a 3-dimensional stratum ℳ3 , V of the moduli space ℳ3 of curves of genus 3 parameterizing curves Y that admit a certain action of V = C2 × C2. We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for ℳ3 , V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (respectively the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants.</p
Wild Galois representations: a family of hyperelliptic curves with large inertia image
In this work we generalise the main result of arXiv:1812.05651 to the family
of hyperelliptic curves with potentially good reduction over a -adic field
which have degree and the largest possible image of inertia under the
-adic Galois representation associated to its Jacobian. We will prove
that this Galois representation factors as the tensor product of an unramified
character and an irreducible representation of a finite group, which can be
either equal to the inertia image (in which case the representation is easily
determined) or a -extension of it. In this second case, there are two
suitable representations and we will describe the Galois action explicitly in
order to determine the correct one.Comment: Keywords: hyperelliptic curves, Galois representations, local fields.
To appear in Mathematical Proceedings of Cambridge Mathematical Societ