Arguments des unit\'es de Stark et p\'eriodes de s\'eries d'Eisenstein

TL;DR

The paper proposes a conjectural method to construct special units in certain non-totally real abelian extensions using topological cycles and Eisenstein series, strengthening Stark's conjecture with numerical evidence.

Contribution

It introduces a new conjectural construction of units in ATR extensions via topological cycles and Eisenstein series, extending Stark's conjecture with explicit argument formulas.

Findings

Conjectural construction aligns with Stark's conjecture for ATR extensions.

Provides an analytic formula for arguments of Stark units.

Numerical evidence supports the conjecture.

Abstract

We describe a conjectural construction (in the spirit of Hilbert's 12th problem) of units in abelian extensions of certain base fields which are neither totally real nor CM. These base fields are quadratic extensions with exactly one complex place of a totally real number field F, and are referred to as Almost Totally real (ATR) extensions. Our construction involves certain null-homologous topological cycles on the Hilbert modular variety attached to F. The special units are the images of these cycles under a map defined by integration of weight two Eisenstein series on GL_2(F). This map is formally analogous to the higher Abel-Jacobi maps that arise in the theory of algebraic cycles. We show that our conjecture is compatible with Stark's conjecture for ATR extensions; it is, however, a genuine strengthening of Stark's conjecture in this context since it gives an analytic formula for the arguments of the Stark units and not just for their absolute values. The last section provides numerical evidences for our conjecture.