Étale Cohomology, the Weil Conjectures, and Deligne¶

Job: develop the étale / $\ell$-adic cohomology (comparison / finiteness), the Weil conjectures (Dwork / Grothendieck / Deligne), the Lefschetz trace formula / point counting, the Deligne Riemann hypothesis / weights / Weil II, and the crystalline / $p$-adic comparison / prismatic — with the Weil / Deligne RH for the elliptic curve / Hasse bound as the worked example.

Core¶

Étale cohomology and the $\ell$-adic cohomology¶

The étale cohomology $H^i_{\mathrm{\acute{e}t}}(X_{\overline{k}},\mathbb{Q}_\ell)$ (the Grothendieck: the sheaf cohomology on the étale site, the $\ell$-adic the inverse limit $\mathbb{Z}/\ell^n$, the finite-dimensional $\mathbb{Q}_\ell$-vector space).

Étale / $\ell$-adic	Description
$H^i_{\mathrm{\acute{e}t}}(X_{\overline{k}},\mathbb{Q}_\ell)$	the Grothendieck étale cohomology (the sheaf cohomology on the étale site)
Comparison (Artin–Grothendieck)	over $\mathbb{C}$ the $H^i_{\mathrm{\acute{e}t}}\otimes\mathbb{C}\cong H^i_{\mathrm{Betti}}$ (the Betti / the de Rham / the Hodge)
Finiteness / constructibility	the proper base change, the finiteness, the constructible sheaves

The Weil conjectures and the zeta function¶

The Weil conjectures (the rationality of $Z(X,t)$, the functional equation, the Riemann hypothesis $|\alpha|=q^{i/2}$).

Weil / Dwork–Grothendieck–Deligne	Description
Weil conjectures	the rationality of $Z(X,t)$, the functional equation, the RH $
Dwork	the $p$-adic proof of the rationality
Grothendieck	the étale cohomology / the Lefschetz trace formula: $Z(X,t)=\prod_i\det(1-\mathrm{Frob}\,t
Deligne	the Riemann hypothesis: $

The Lefschetz trace formula and the point counting¶

Lefschetz / point counting	Description
Lefschetz trace	$#X(\mathbb{F}_{q^n})=\sum_i(-1)^i\mathrm{Tr}(\mathrm{Frob}^n
Grothendieck–Lefschetz	the fixed-point formula (the trace, the $\mathrm{Frob}$ the fixed points)
Deligne weights	the weights of the Frobenius eigenvalues (the weight filtration, the mixed / the pure)

The Riemann hypothesis (Deligne) and the weights¶

Theorem (Deligne, 1974). The Riemann hypothesis for varieties over finite fields: $|\alpha|=q^{i/2}$, the eigenvalues of the Frobenius on $H^i$ of absolute value $q^{i/2}$.

Deligne RH / weights	Description
Deligne RH	$
Weights	the $i$-th cohomology the weight $i$ (the pure / the mixed, the Deligne the weight filtration)
Weil II	the Deligne: the weights / the Fourier–Deligne, the sheaves the weights

The crystalline cohomology and the $p$-adic¶

Crystalline / $p$-adic / prismatic	Description
Crystalline (Berthelot)	the $p$-adic cohomology for the smooth / proper over $\mathbb{F}_p$
$p$-adic comparison (Fontaine)	the crystalline / the de Rham / the étale, the $p$-adic Hodge the comparison
Frobenius / slopes (Dieudonné–Manin)	the slopes, the $p$-adic Hodge the slope decomposition
Prismatic (Bhatt–Scholze)	the prismatic cohomology the generalization (the prismatic)

See it¶

Weil conjectures for an elliptic curve over $\mathbb{F}_q$ and the Deligne Riemann hypothesis. For the elliptic curve $E/\mathbb{F}_q$ (the smooth projective genus 1), the zeta $$ Z(E,t)=\frac{(1-\alpha t)(1-\bar\alpha t)}{(1-t)(1-qt)} $$ (the $H^1$ the 2-dimensional, the $\alpha,\bar\alpha$ the Frobenius eigenvalues on $H^1_{\mathrm{\acute{e}t}}$). The Riemann hypothesis: $|\alpha|=|\bar\alpha|=q^{1/2}$ (the Deligne, the $H^1$ the weight 1). The point counting: $\#E(\mathbb{F}_{q^n})=q^n+1-\alpha^n-\bar\alpha^n$ (the Lefschetz trace, the $\mathrm{Tr}(\mathrm{Frob}^n|H^0)-\mathrm{Tr}(\mathrm{Frob}^n|H^1)+\mathrm{Tr}(\mathrm{Frob}^n|H^2)=1-\alpha^n-\bar\alpha^n+q^n$). The Hasse: $|\#E(\mathbb{F}_{q^n})-(q^n+1)|\le2q^{n/2}$ (the Hasse bound, the $|\alpha|=q^{1/2}$ the RH). The key point: the Weil / the Deligne RH for the elliptic curve — the $|\alpha|=q^{1/2}$ (the weight 1), the point counting via the Lefschetz trace, the Hasse bound the consequence — the étale cohomology / the Frobenius the arithmetic.

Check¶

Verify the Lefschetz trace formula for the elliptic curve $E/\mathbb{F}_q$: $\#E(\mathbb{F}_{q^n})=q^n+1-\alpha^n-\bar\alpha^n$, and the Hasse bound.

The Lefschetz trace formula: $\#E(\mathbb{F}_{q^n})=\sum_i(-1)^i\mathrm{Tr}(\mathrm{Frob}^n|H^i_{\mathrm{\acute{e}t}})$. For the elliptic curve $E/\mathbb{F}_q$ (the smooth projective genus 1), $H^0_{\mathrm{\acute{e}t}}=\mathbb{Q}_\ell$ (the $\mathrm{Tr}(\mathrm{Frob}^n|H^0)=1$, the Frob the identity on $H^0$), $H^1_{\mathrm{\acute{e}t}}=\mathbb{Q}_\ell^2$ (the $\mathrm{Tr}(\mathrm{Frob}^n|H^1)=\alpha^n+\bar\alpha^n$, the $\alpha,\bar\alpha$ the eigenvalues), $H^2_{\mathrm{\acute{e}t}}=\mathbb{Q}_\ell(-1)$ (the $\mathrm{Tr}(\mathrm{Frob}^n|H^2)=q^n$, the $\mathbb{Q}_\ell(-1)$ the Tate twist, the Frob $q$). So $\#E(\mathbb{F}_{q^n})=1-(\alpha^n+\bar\alpha^n)+q^n=q^n+1-\alpha^n-\bar\alpha^n$. The Hasse bound: $|\#E(\mathbb{F}_{q^n})-(q^n+1)|=|\alpha^n+\bar\alpha^n|\le2|\alpha|^n=2q^{n/2}$ (the $|\alpha|=q^{1/2}$ the RH, the $|\alpha^n+\bar\alpha^n|\le2|\alpha|^n$). So the Hasse bound $|\#E-(q+1)|\le2q^{1/2}$ the consequence of the RH.

Explain the Deligne weights and the pure vs mixed, and identify the weight filtration.

The Deligne RH: the eigenvalues of the Frobenius on $H^i_{\mathrm{\acute{e}t}}(X_{\overline{\mathbb{F}}_q},\mathbb{Q}_\ell)$ have the absolute value $q^{i/2}$ (the weight $i$, the pure). The pure: the $X$ smooth projective $\Rightarrow$ the $H^i$ the pure weight $i$ (all eigenvalues $|\alpha|=q^{i/2}$). The mixed: the $X$ non-smooth / the open / the singular $\Rightarrow$ the $H^i$ the mixed (the weights $\le i$, the weight filtration $W_\bullet$ the Deligne). The weight filtration $W_\bullet H^i$ (the increasing filtration, the $\mathrm{Gr}^W_j H^i$ the pure weight $j$, the Deligne the mixed Hodge / the weights). The Weil II (the Deligne): the sheaves the weights (the Fourier–Deligne, the weights of the sheaves, the lisse / the constructible), the pure / the mixed the weights.

Pitfalls¶

The étale / the $\ell\ne p$ (the $p$-adic the crystalline / the prismatic, the $\ell$-adic the étale, the two the different). The étale cohomology the $\ell\ne\mathrm{char}\,k$, the $p$-adic the crystalline / the prismatic (the $p=\mathrm{char}$), the two the different cohomologies.
The Deligne RH the weights (the $|\alpha|=q^{i/2}$ the weight $i$, the pure / the mixed — the mixed the non-smooth). The pure the smooth projective, the mixed the non-smooth / the open.
The Lefschetz trace / the point counting the $\#X(\mathbb{F}_{q^n})$ via the $\mathrm{Tr}(\mathrm{Frob}^n|H^i)$ (the étale the arithmetic), not the de Rham / the Betti (the étale the finite field, the Betti the complex).
The crystalline / the $p$-adic the Berthelot (the $p$-adic the smooth / proper over $\mathbb{F}_p$), the Fontaine the comparison (the crystalline / the de Rham / the étale), the prismatic the Bhatt–Scholze the generalization (the prismatic, the recent).
The Dieudonné–Manin / the slopes the $p$-adic Hodge the slope decomposition (the Frobenius the slopes, the $p$-adic), the Fontaine the rings $B_{\mathrm{cris}},B_{\mathrm{st}},B_{\mathrm{dR}}$ (the $p$-adic Hodge, the local).

Refs¶

Grothendieck, SGA 4–4½ (the étale cohomology); Artin–Grothendieck, the comparison
Deligne, "La conjecture de Weil I–II" (Publ. Math. IHÉS 1974, 1980); the RH / the weights / the Weil II
Dwork, "On the rationality of the zeta function" (1960); the $p$-adic rationality
Berthelot, Cohomologie cristalline (1974); Fontaine, the $p$-adic Hodge; Bhatt–Scholze, "Prisms" (Annals 2022); Dieudonné–Manin, the slopes

Next¶

AG index · Foundations · Mordell–Weil & BSD · Diophantine & Arakelov · Galois Reps & Fontaine–Mazur · Prismatic cohomology · AG cohomology