Mathematical Research Letters

Volume 28 (2021)

Number 5

Irreducibility of geometric Galois representations and the Tate conjecture for a family of elliptic surfaces

Pages: 1353 – 1378



Lian Duan (Department of Mathematics, Colorado State University, Fort Collins, Colo., U.S.A.)

Xiyuan Wang (Department of Mathematics, Johns Hopkins University, Baltimore, Maryland, U.S.A.)


Using Calegari’s result on the Fontaine–Mazur conjecture, we study the irreducibility of pure, regular, rank $3$ weakly compatible systems of self-dual $\ell$-adic representations. As a consequence, we prove that the Tate conjecture holds for a family of elliptic surfaces defined over $\mathbf{Q}$ with geometric genus bigger than $1$.

Received 27 April 2020

Accepted 25 August 2020

Published 16 August 2022