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• Tags
  • #arithmetic-geometry
• Refs: #resources
  • RH over finite fields:
    • attachments/1509.00797.pdf
  • AG book which includes the Hasse-Weil inequality:
    • attachments/ag.pdf
  • RH for curves and hypersurfaces over finite fields:
    • attachments/baby16.pdf
  • Computing Zeta functions:
    • Computing Zeta Functions.pdf
    • Computing Zeta over FF.pdf
  • Weil’s paper:
    • attachments/euclid.bams.1183513798.pdf
  • Gauss and Jacobi sums:
    • Gauss Jacobi Sums.pdf
  • General notes:
    • weil.pdf
  • Hypersurfaces and the Weil conjectures:
    • weil-preprint1.pdf
  • The zeta book:
    • zeta_book.pdf
  • Lecture notes with cohomological proofs
  • 2019 Lecture notes on the conjectures: http://pagine.dm.unipi.it/tamas/Weil.pdf#page=2 #resources/course-notes
• Links:
  • Weil cohomology
  • purity theorem
  • Frobenius morphism
  • applications of Weil conjectures
  • l-adic cohomology
  • Grothendieck-Lefschetz Trace Formula
  • Applications and consequences:
    • Ramanujan-Petersson conjecture
    • Generalized estimates for Kloosterman sums.

Weil Conjectures

Notes

2021-04-28_Weil_Conjectures_1

2021-04-28_Weil_Conjectures_2

2021-04-28_Weil_Conjectures_3

2021-04-28_Weil_Conjectures_4

2021-04-28_Weil_Conjectures_Talk

Results

For general projective curves of genus \(g\), need to use the Tate module of the Jacobian. Result:

Statement of Weil Conjectures

Point counts using traces

Proofs

Dwork

Rationality criteria

See also showing an analytic function is rational

See determinant of an operator

Induction by hyperplane slicing

Character sums

## Deligne

See https://people.math.ethz.ch/~kowalski/deligne.pdf#page=19

Reductions and weak Lefschetz:

Reduction to an inequality:

Hyperplane slicing step

Deligne’s proof

See http://www.mathematik.uni-regensburg.de/Jannsen/home/Weil-gesamt-eng.pdf#page=74

History of proofs

• F. K. Schmidt proved these statements for curves, except for the Riemann hypothesis part, which was proved by Hasse for elliptic curves and Weil for arbitrary curves.

• Weil proved these statements also for abelian varieties.

• Dwork used p-adic analysis proves part (i) for varieties of arbitrary dimension, without finding a cohomology theory.

• Grothendieck partially using Grothendieck-Lefschetz Trace Formula:

  \begin{align*} {\sharp}X\left(\mathbb{F}_{q}\right)=\sum_{i=0}^{2 d}(-1)^{i} \operatorname{Tr}\left(\left.F^{*}\right|_{H_{\mathrm{et}}^{i}\left(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu, \mathbb{Q}_{\ell}\right)}\right) \end{align*}

• RH proved by Deligne.

• Led to development of etale cohomology by Grothendieck and M. Artin

Proof of Rationality

Refined proof of rationality

There are two obvious conjectures, and one is significantly stronger than the other. We could ask whether \(\zeta\left(X_{0}, t\right)\) has rational coefficients, or more generally whether each individual \(P_{i}(t)\) has rational coefficients and is independent of the choice of \(l\). We are now in a position where proof of the former is relatively straightforward, and the latter statement can be reduced to a statement about absolute values of eigenvalues.

Proof of Duality

Proof of RH

See Euler Product Expansion

Remarks about RH

For curves

Zeta functions

Recovers Dedekind zeta function:

Misc

See reduced schemes of finite type, closed point, degree of a closed point. Frobenius morphism.

motivic behavior of zeta functions:

Example computations

Recovering number of points from rational expression