Jerome William Hoffman

arXiv2024

QM abelian varieties, hypergeometric character sums and modular forms

This is a report on recent work, with Wen-Ching Winnie Li and Ling Long. In that work explicit formulas are given, involving hypergeometric character sums, for the traces of Hecke operators $T_p$ acting spaces of cusp forms $S_k(Γ)$ of weight $k$ for certain arithmetically defined Fuchsian subgroups $Γ\subset \mathrm{SL}_2 (\mathbf{R})$. In particular we consider the groups attached to the quaternion division algebra $B_6$ over $\mathbf{Q}$ of discriminant 6.

math.NT↗

arXiv2015

Infinitesimal Theory of Chow Groups via K-theory, Cyclic Homology and the Relative Chern Character

We examine the tangent groups at the identity, and more generally the formal completions at the identity, of the Chow groups of algebraic cycles on a nonsingular quasiprojective algebraic variety over a field of characteristic zero. We settle a question recently raised by Mark Green and Phillip Griffiths concerning the existence of Bloch-Gersten-Quillen-type resolutions of algebraic K-theory sheaves on infinitesimal thickenings of nonsingular varieties, and the relationships between these sequences and their "tangent sequences," expressed in terms of absolute Kähler differentials. More generally, we place Green and Griffiths' concrete geometric approach to the infinitesimal theory of Chow groups in a natural and formally rigorous structural context, expressed in terms of nonconnective K-theory, negative cyclic homology, and the relative algebraic Chern character

math.AG↗

arXiv2007

Picard groups of Siegel modular threefolds and theta lifting

We show that the Humbert surfaces rationally generate the Picard groups of Siegel modular threefolds.

math.AG↗