arXiv subjects

Stephen Lichtenbaum

Publications and source records attributed to Stephen Lichtenbaum.

3 recordsLinked to original sources

arXiv2021

Special Values of Zeta Functions of Schemes

Let X be a regular scheme, projective and flat over Spec \mathbb Z. We give a conjectural formula, up to sign and powers of 2, for ζ^*(X,r), the leading term in the series expansion of ζ(X,s) at s=r, in terms of Weil-etale motivic cohomology, singular cohomology, and derived de Rham cohomology. This formula builds on work of Fontaine and Perrin-Riou replacing \mathbb Q_structuers by \mathbb Z-structures. We show that our conjectured formula is compatible with Serre's functional equation for the zeta function of X.

math.AG↗

arXiv2018

The constant in the functional equation and derived extrior powers

Let X be a regular scheme, projective and flat over the integers. Let A be the constant in the conjectured functional equation for the zeta-function of X. We give a conjecture computing A in terms of Euler characteristics of derived exterior powers of the sheaf of Kahler differentials on X, and prove this conjecture when the dimension of X is 1 or 2. This conjecture essentially says that the formulas for the special values of the zeta-function of X given by the author in a previous preprint are compatible with the functional equation.

math.AG↗

arXiv2005

The Weil-Etale Topology for Number Rings

We would like to construct a new Grothendieck topology for arithmetic schemes, whose cohomology groups associated with motivic complexes of sheaves are finitely generated and whose Euler characteristics are related to special values of zeta-functions. In this paper we construct this topology for rings of algebraic integers and show that the cohomology of the constant sheaf Z is related to the behavior of zeta-functions at s = 0.

math.NT↗