--- date: 2021-10-25 23:31 modification date: Tuesday 26th October 2021 00:14:29 title: Weil Conjectures aliases: [Weil Conjectures] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #arithmetic-geometry - Refs: #resources - RH over finite fields: - [attachments/1509.00797.pdf](attachments/1509.00797.pdf) - AG book which includes the Hasse-Weil inequality: - [attachments/ag.pdf](attachments/ag.pdf) - RH for curves and hypersurfaces over finite fields: - [attachments/baby16.pdf](attachments/baby16.pdf) - Computing Zeta functions: - [Computing Zeta Functions.pdf](Computing%20Zeta%20Functions.pdf) - [Computing Zeta over FF.pdf](Computing%20Zeta%20over%20FF.pdf) - Weil's paper: - [attachments/euclid.bams.1183513798.pdf](attachments/euclid.bams.1183513798.pdf) - Gauss and Jacobi sums: - [Gauss Jacobi Sums.pdf](Gauss%20Jacobi%20Sums.pdf) - General notes: - [weil.pdf](weil.pdf) - Hypersurfaces and the Weil conjectures: - [weil-preprint1.pdf](weil-preprint1.pdf) - The zeta book: - [zeta_book.pdf](zeta_book.pdf) - [Lecture notes with cohomological proofs](https://people.math.harvard.edu/~mpopa/571/chapter2.pdf) - 2019 Lecture notes on the conjectures: #resources/course-notes - Links: - [Weil cohomology](Weil%20cohomology.md) - [purity theorem](Unsorted/purity%20theorem.md) - [Frobenius morphism](Frobenius%20morphism.md) - [applications of Weil conjectures](applications%20of%20Weil%20conjectures.md) - [l-adic cohomology](Unsorted/l-adic%20cohomology.md) - [[Unsorted/Grothendieck-Lefschetz Trace Formula]] - Applications and consequences: - [Ramanujan-Petersson conjecture](Unsorted/Ramanujan-Petersson%20conjecture.md) - Generalized estimates for [Kloosterman sums](Kloosterman%20sums). --- # Weil Conjectures ## Notes [2021-04-28_Weil_Conjectures_1](2021-04-28_Weil_Conjectures_1.md) [2021-04-28_Weil_Conjectures_2](2021-04-28_Weil_Conjectures_2.md) [2021-04-28_Weil_Conjectures_3](2021-04-28_Weil_Conjectures_3.md) [2021-04-28_Weil_Conjectures_4](2021-04-28_Weil_Conjectures_4.md) [2021-04-28_Weil_Conjectures_Talk](2021-04-28_Weil_Conjectures_Talk.md) # Results ![](attachments/Pasted%20image%2020220207122124.png) For general projective curves of genus $g$, need to use the [Tate module](Unsorted/Tate%20module.md) of the [Jacobian](Jacobian). Result: ![](attachments/Pasted%20image%2020220207122232.png) # Statement of Weil Conjectures ![](attachments/Pasted%20image%2020220207122241.png) ![](attachments/Pasted%20image%2020220207122317.png) ![](attachments/Pasted%20image%2020220214113259.png) ![](attachments/Pasted%20image%2020220214114204.png) ![](attachments/Pasted%20image%2020220214114214.png) ![](attachments/Pasted%20image%2020220214113324.png) ![](attachments/Pasted%20image%2020220214113611.png) ![](attachments/Pasted%20image%2020220510170517.png) # Point counts using traces ![](attachments/Pasted%20image%2020220424123435.png) ![](attachments/Pasted%20image%2020220424123512.png) ![](attachments/Pasted%20image%2020220424123709.png) # Proofs ## Dwork ![](attachments/Pasted%20image%2020220510170558.png) ![](attachments/Pasted%20image%2020220510170636.png) ### Rationality criteria See also [showing an analytic function is rational](showing%20an%20analytic%20function%20is%20rational.md) ![](attachments/Pasted%20image%2020220510170728.png) ![](attachments/Pasted%20image%2020220510171219.png) ![](attachments/Pasted%20image%2020220510171245.png) ![](attachments/Pasted%20image%2020220510171411.png) ![](attachments/Pasted%20image%2020220510171444.png) See [determinant of an operator](determinant%20of%20an%20operator.md) ## Induction by hyperplane slicing ![](attachments/Pasted%20image%2020220510171834.png) ## Character sums ![](attachments/Pasted%20image%2020220510171912.png) ![](attachments/Pasted%20image%2020220510172049.png) ## Deligne See ![](attachments/Pasted%20image%2020220424130753.png) Reductions and [weak Lefschetz](Unsorted/weak%20and%20hard%20Lefschetz%20theorems.md): ![](attachments/Pasted%20image%2020220424131501.png) Reduction to an inequality: ![](attachments/Pasted%20image%2020220424131635.png) ![](attachments/Pasted%20image%2020220424131648.png) ## Hyperplane slicing step ![](attachments/Pasted%20image%2020220424132013.png) ![](attachments/Pasted%20image%2020220424132112.png) ![](attachments/Pasted%20image%2020220424132433.png) ![](attachments/Pasted%20image%2020220424133305.png) ![](attachments/Pasted%20image%2020220424133617.png) ![](attachments/Pasted%20image%2020220424133639.png) ![](attachments/Pasted%20image%2020220424133700.png) ![](attachments/Pasted%20image%2020220424133932.png) ![](attachments/Pasted%20image%2020220424133954.png) ![](attachments/Pasted%20image%2020220424134014.png) ![](attachments/Pasted%20image%2020220424134033.png) ![](attachments/Pasted%20image%2020220424134125.png) ![](attachments/Pasted%20image%2020220424134105.png) ![](attachments/Pasted%20image%2020220424134218.png) ## Deligne's proof See ![](attachments/Pasted%20image%2020220424143244.png) # 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]]: \[ \size X\left(\mathbb{F}_{q}\right)=\sum_{i=0}^{2 d}(-1)^{i} \operatorname{Tr}\left(\left.F^{*}\right|_{H_{\mathrm{et}}^{i}\left(\bar{X}, \mathbb{Q}_{\ell}\right)}\right) \] - RH proved by Deligne. - Led to development of etale cohomology by Grothendieck and M. Artin ## Proof of Rationality ![](attachments/Pasted%20image%2020220207122507.png) ### 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. ![](attachments/Pasted%20image%2020220207122942.png) ## Proof of Duality ![](attachments/Pasted%20image%2020220207122802.png) ## Proof of RH ![](attachments/Pasted%20image%2020220207123022.png) ![](attachments/Pasted%20image%2020220207123253.png) See [Euler Product Expansion](Unsorted/Riemann%20Zeta.md#Euler%20Product%20Expansion) # Remarks about RH ![](attachments/Pasted%20image%2020220424122340.png) ![](attachments/Pasted%20image%2020220424122352.png) ![](attachments/Pasted%20image%2020220424122402.png) # For curves ![](attachments/Pasted%20image%2020220214113000.png) # Zeta functions ![](attachments/Pasted%20image%2020220214114050.png) ![](attachments/Pasted%20image%2020220214114106.png) Recovers [Dedekind zeta function](Unsorted/Dedekind%20zeta%20function.md): ![](attachments/Pasted%20image%2020220214114133.png) ![](attachments/Pasted%20image%2020220214114331.png) ![](attachments/Pasted%20image%2020220217213222.png) # Misc See [reduced](Unsorted/reduced.md) schemes of [[finite type]], [closed point](Unsorted/special%20fiber.md), [degree of a closed point](degree%20of%20a%20closed%20point.md). [Frobenius morphism](Unsorted/Frobenius%20morphism.md). ![](attachments/Pasted%20image%2020220401101552.png) [motivic](Unsorted/motive%20MOC.md) behavior of zeta functions: ![](attachments/Pasted%20image%2020220401101656.png) # Example computations ![](attachments/Pasted%20image%2020220401101837.png) ![](attachments/Pasted%20image%2020220401101849.png) # Recovering number of points from rational expression ![](attachments/Pasted%20image%2020220410153813.png)