The Tate Conjecture for a family of surfaces of general type with p_g=q=1 and K^2=3

TL;DR

This paper proves the Tate conjecture for a specific family of complex algebraic surfaces of general type with invariants p_g=q=1 and K^2=3, using monodromy and degeneration techniques.

Contribution

It establishes the Tate conjecture for this family by demonstrating big monodromy and analyzing degenerations, advancing understanding of algebraic surfaces with these invariants.

Findings

Verified the semisimplicity of the Galois representation.

Confirmed the Tate conjecture for surfaces over finitely generated fields.

Proved a big monodromy result for the family of surfaces.

Abstract

We prove a big monodromy result for a smooth family of complex algebraic surfaces of general type, with invariants p_g=q=1 and K^2=3, that has been introduced by Catanese and Ciliberto. This is accomplished via a careful study of degenerations. As corollaries, when a surface in this family is defined over a finitely generated extension of Q, we verify the semisimplicity and Tate conjectures for the Galois representation on the middle \ell-adic cohomology of the surface.