The non-existence of some Galois representations of moderate dimension in small characteristic

2025/08/30 by Alexandru Ghitza, Ghitza, Alexandru, Takuya Yamauchi +1
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2509.00635

Abstract

Refining arguments of Hyunsuk Moon, under the assumption of the Generalized Riemann Hypothesis, we prove the non-existence of irreducible mod 2 Galois representations unramified outside 2 of dimensions ≤ 4, and of totally real such representations of dimensions ≤ 8. We also prove the non-existence of irreducible totally real mod 3 representations unramified outside 3 of dimensions ≤ 4. We show unconditionally that the image of an irreducible mod 2 symplectic 4-dimensional Galois representation that is unramified outside 2 must be large. Under GRH, we then deduce the non-existence of such representations.

Citations

• Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
• Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials
• Ramification in the Inverse Galois Problem
• An equidistribution theorem for holomorphic Siegel modular forms for GSp₄
• Galois extensions ramified only at one prime
• The non-existence of certain mod 2 Galois representations of some small quadratic fields

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