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Weil conjectures and affine hypersurfaces

Dingxin Zhang

TL;DR

The paper offers an alternative proof of Deligne's Riemann Hypothesis for smooth projective varieties over a finite field by reducing to the hypersurface case through deformation to affine hypersurfaces, aided by Artin's vanishing and perverse sheaf technology. Central to the approach are the perverse degeneration lemma and perverse t-structure results, which enable precise control of Frobenius weights across families and degenerations. By connecting weight bounds to zeta-function behavior via Grothendieck's trace formula and employing Katz's hypersurface results, the authors deduce weight $\le i/2$ on $H^i_c$ and hence establish Deligne's RH for all smooth proper varieties. The method complements earlier reductions (e.g., Scholl) and provides a relatively streamlined, deformation-based route with Weil II principles at its core, potentially broadening the toolkit for understanding arithmetic and geometric properties of varieties over finite fields.

Abstract

We give yet another proof of the Riemann hypothesis for smooth projective varieties over a finite field (Deligne's theorem), by reducing to the hypersurface case. The latter was established by N. Katz via an elementary argument. A reduction of this kind was previously carried out by A. J. Scholl. Our approach is slightly different, and relies on deformation to an affine hypersurface, together with Artin's vanishing theorem and basic properties of perverse sheaves.

Weil conjectures and affine hypersurfaces

TL;DR

The paper offers an alternative proof of Deligne's Riemann Hypothesis for smooth projective varieties over a finite field by reducing to the hypersurface case through deformation to affine hypersurfaces, aided by Artin's vanishing and perverse sheaf technology. Central to the approach are the perverse degeneration lemma and perverse t-structure results, which enable precise control of Frobenius weights across families and degenerations. By connecting weight bounds to zeta-function behavior via Grothendieck's trace formula and employing Katz's hypersurface results, the authors deduce weight on and hence establish Deligne's RH for all smooth proper varieties. The method complements earlier reductions (e.g., Scholl) and provides a relatively streamlined, deformation-based route with Weil II principles at its core, potentially broadening the toolkit for understanding arithmetic and geometric properties of varieties over finite fields.

Abstract

We give yet another proof of the Riemann hypothesis for smooth projective varieties over a finite field (Deligne's theorem), by reducing to the hypersurface case. The latter was established by N. Katz via an elementary argument. A reduction of this kind was previously carried out by A. J. Scholl. Our approach is slightly different, and relies on deformation to an affine hypersurface, together with Artin's vanishing theorem and basic properties of perverse sheaves.
Paper Structure (5 sections, 11 theorems, 44 equations)

This paper contains 5 sections, 11 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. The perverse degeneration lemma
  3. A few facts about the perverse t-structure
  4. Trivial bounds and semicontinuity of weights
  5. Proof of Theorem \ref{['theorem:wrh']}

Key Result

Theorem 1.1

Assume $X_0$ is smooth and proper over $\mathbf{F}_q$. Suppose $\alpha$ is an eigenvalue of the geometric Frobenius $F$ acting on $\mathrm{H}^i_{c}(X;\overline{\mathbf{Q}}_{\ell})=\mathrm{H}^i(X;\overline{\mathbf{Q}}_{\ell})$. Then for any isomorphism $\iota\colon \overline{\mathbf{Q}}_{\ell} \to \m

Theorems & Definitions (30)

  • Theorem 1.1: Deligne deligne_weil1
  • Definition 1.2: Weight
  • Theorem 1.3
  • Theorem 1.4
  • proof : Proof that Theorem \ref{['theorem:wrh']} $\Rightarrow$ Theorem \ref{['theorem:rh']}
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6: cf. katz_perverse-origin, wan-zhang_betti-number-bounds-for-varieties-and-exponential-sums0
  • ...and 20 more