We provide proofs of Theorems 4.1 and 5.1 of Khare and Wintenberger (Invent. Math., doi:10.1007/s00222-009-0205-7, 2009).

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Serre’s modularity conjecture (II)

This paper is the first part of a work which proves Serre’s modularity conjecture. We first prove the cases p 6= 2 and odd conductor, see Theorem 1.2, modulo Theorems 4.1 and 5.1. Theorems 4.1 and 5.1 are proven in the second part, see [18]. We then reduce the general case to a modularity statement for 2-adic lifts of modular mod 2 representations. This statement is now a theorem of Kisin [24].

Serre's Modularity Conjecture

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Lectures on Serre’s conjectures

We prove a numerical form of a Jacquet–Langlands correspondence for torsion classes on arithmetic hyperbolic 3-manifolds.

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A Torsion Jacquet-langlands Correspondence

We study torsion in the homology of arithmetic groups and give evidence that it plays a role in the Langlands program. We prove, among other results, a numerical form of a Jacquet--Langlands correspondence in the torsion setting.

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A torsion Jacquet--Langlands correspondence

Finiteness Results on Certain mod p Galois Representations

Non-existence is proved of certain continuous irreducible mod p representations of degree 2 of the absolute Galois group of the rational number field. This extends previously known results, the improvement based on a refinement of Tate’s discriminant bound. 2000 Mathematics Subject Classification: 11F80, 11R29, 11R39

Refinement of Tate's discriminant bound and non-existence theorems for mod Galois representations.

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Mod p Galois representations of solvable image

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The number of monomial mod p Galois representations with bounded conductor

The non-existence is proved of continuous irreducible representations p : Gal(Q/Q) \longrightarrow GL(F) with Artin conductor N outside p for a few small values of p and N.N.

Hyunsuk Moon

Papers

Finiteness Results on Certain mod p Galois Representations

Refinement of Tate's discriminant bound and non-existence theorems for mod Galois representations.

Mod p Galois representations of solvable image

The number of monomial mod p Galois representations with bounded conductor

THE NON-EXISTENCE OF CERTAIN MOD p GALOIS REPRESENTATIONS