Wild Galois representations: elliptic curves with wild cyclic reduction

Area of Science:

• Number Theory
• Algebraic Geometry
• Galois Representations

Background:

• Kraus classified inertia images of l-adic Galois representations for elliptic curves over non-archimedean fields in 1990.
• Previous work computed Galois representations for elliptic curves with non-abelian inertia images, occurring when residue characteristic is 2 or 3.

Purpose of the Study:

• To complete the computation of Galois representations in remaining "wild" cases.
• To analyze elliptic curves with residue characteristic p=2 or 3 and good reduction over extensions with p-divisible ramification degree.

Main Methods:

• Explicit computation of Galois representations.
• Analysis of elliptic curves over non-archimedean local fields.
• Building upon previous classifications and computations of inertia images.

Main Results:

• The computation of Galois representations is completed for all remaining wild cases.
• This work addresses scenarios where the ramification degree is divisible by the residue characteristic (p=2 or 3).
• The condition of a non-abelian inertia image is not assumed, broadening the scope of analysis.

Conclusions:

• The study provides a comprehensive understanding of Galois representations for elliptic curves in specific wild ramification scenarios.
• This research contributes to the explicit computation of Galois representations in algebraic number theory.
• The findings are based on Chapter V of the author's PhD thesis.