The non-existence of some Galois representations of moderate dimension in small characteristic
Alexandru Ghitza, Takuya Yamauchi
TL;DR
This work tackles the non-existence of low-dimensional mod-$p$ Galois representations with small ramification. It combines two routes: under GRH, refinements of Moon’s discriminant-based approach rule out irreducible representations of dimensions up to $4$ (and certain totally real cases up to dimension $8$) for $p=2$ and $p=3$; and unconditionally it proves large-image phenomena for $2$-ramified $GSp_4$-valued representations, with GRH then extending non-existence to these symplectic cases. The analysis hinges on sharpened discriminant bounds tied to $p$-lengths, Brauer-type counting of modular representations, and explicit finite-group classifications (notably within $\mathrm{GSp}_4$ and related subgroups), supplemented by computational checks (Magma/Sage) to exhaust remaining possibilities. The results constrain potential connections to automorphy or modularity for small-dimensional Galois representations and illustrate the power of combining analytic GRH-based methods with exact group-theoretic classifications. Overall, the paper advances our understanding of how small ramification restricts the landscape of mod-$p$ Galois representations, with implications for Serre-type modularity questions.
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 $\leq 4$, and of totally real such representations of dimensions $\leq 8$. We also prove the non-existence of irreducible totally real mod 3 representations unramified outside 3 of dimensions $\leq 4$. 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.