Moscow Mathematical Journal

Volume 25, Issue 3, July–September 2025 pp. 389–416.

Group Cohomology and Lattice Invariants over Real Quadratic Fields

Authors: Milton Espinoza (1)
Author institution:(1) Departamento de Matemáticas, Facultad de Ciencias, Universidad de La Serena, Juan Cisternas 1200, La Serena, Chile

Summary:

We introduce invariants of lattices in real quadratic fields that are constructed from the first derivative at $s=0$ of certain $L$-series. These invariants are able to distinguish the contribution of each of the two embeddings of the base field into $\mathbb{R}$. Our construction makes use of the first cohomology group of $\mathrm{PGL}_2(\mathbb{Q})$ with coefficients in a module of distributions. This technique allows us to control and sometimes remove the effect of choosing coordinates in the description of such lattices. Furthermore, we explicitly compute the invariants in the simplest cases.

2020 Math. Subj. Class. Primary: 11F75, 11M32, 11R42; Secondary: 11R11, 11A55.

Keywords: Partial zeta functions, Stark units, real quadratic fields, $\mathrm{PGL}_2(\mathbb{Q})$, group cocycles, continued fractions.

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