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Trace cohomology revisited

Igor V. Nikolaev

TL;DR

This work introduces trace cohomology, derived from Serre C*-algebras, as a noncommutative-geometric tool to study the zeta functions of Kuga-Sato varieties. By linking trace cohomology to cusp forms through the Deligne–Scholl correspondence and employing the Petersson inner product, it derives an RH-type bound for Frobenius eigenvalues, providing an alternative route to Weil conjectures in this setting. The method yields explicit computations for curves (n=1), including CM elliptic cases, and demonstrates how trace cohomology encodes arithmetic information via a real-valued cohomological embedding. Overall, the paper showcases a novel intersection of noncommutative geometry and number theory with potential to address open questions through a trace-theoretic lens.

Abstract

We use a cohomology theory coming from the canonical trace on a C*-algebra of the projective variety to prove an analog of the Riemann Hypothesis for the Kuga-Sato varieties over finite fields.

Trace cohomology revisited

TL;DR

This work introduces trace cohomology, derived from Serre C*-algebras, as a noncommutative-geometric tool to study the zeta functions of Kuga-Sato varieties. By linking trace cohomology to cusp forms through the Deligne–Scholl correspondence and employing the Petersson inner product, it derives an RH-type bound for Frobenius eigenvalues, providing an alternative route to Weil conjectures in this setting. The method yields explicit computations for curves (n=1), including CM elliptic cases, and demonstrates how trace cohomology encodes arithmetic information via a real-valued cohomological embedding. Overall, the paper showcases a novel intersection of noncommutative geometry and number theory with potential to address open questions through a trace-theoretic lens.

Abstract

We use a cohomology theory coming from the canonical trace on a C*-algebra of the projective variety to prove an analog of the Riemann Hypothesis for the Kuga-Sato varieties over finite fields.

Paper Structure

This paper contains 8 sections, 8 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Serre $C^*$-algebra
  4. Trace cohomology
  5. Weil's Conjectures
  6. Kuga-Sato varieties
  7. Proof of theorem \ref{['thm1']}
  8. Examples

Key Result

Theorem 1.1

The roots $\alpha_{ij}$ of polynomials $P_i(t)$ in the zeta function $Z_V(t)={P_1(t)\dots P_{2n-1}(t)\over P_0(t)\dots P_{2n}(t)}$ of $V({\Bbb F}_q)$ are algebraic numbers of the absolute value $|\alpha_{ij}|=q^{{i\over 2}}$.

Figures (1)

  • Figure 1: Trace cohomology.

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 3.1
  • proof
  • ...and 18 more