Heisenberg representations and indecomposable division algebras

Abstract: Let $F$ be a field, with absolute Galois group G. Let $p$ be a prime. Denote by $B_d$ the Borel subgroup of $GL_d$, and by $U_d$ its unipotent radical. We consider the question of lifting a triangular Galois representation $G \longrightarrow B_d(\mathbb Z/p)$, to its mod $p^2$ analogue $G \longrightarrow B_d(\mathbb Z/p^2)$. It has a rich history, which we will recall. We'll then explain positive results, up to $d=3$, under the presence of $p^2$-th root of unity in $F$. Using an indecomposability result for divisions algebras, due to Karpenko, we'll show that the answer to the analogous question, with $U_3$ in place of $B_3$, is negative.

algebraic geometrygroup theorynumber theoryrepresentation theory

Audience: researchers in the topic

Algebraic groups and algebraic geometry: in honor of Zinovy Reichstein's 60th birthday