Arithmetic Geometry: Foundations¶

Job: set varieties over number fields and finite fields; rational points and heights; state Faltings’ theorem; point to étale cohomology as the source of Galois representations.

Core¶

Varieties over global and finite fields¶

Let $K$ be a number field or $K=\mathbb{F}_q(t)$ (global function field). A variety over $K$ is a separated scheme of finite type over $K$ (often geometrically integral, smooth, proper when stated). An integral model over $\mathcal{O}_{K,S}$ extends $X/K$ to a scheme $\mathcal{X}/\mathcal{O}_{K,S}$; fibers $\mathcal{X}_{\mathbb{F}_v}$ are reductions at places $v\notin S$.

Base	Typical questions
$\mathbb{Q}$ or $K$	$X(K)$ finite/infinite? density? Hasse principle?
$\mathbb{F}_q$	$\#X(\mathbb{F}_{q^n})$; zeta function; Weil conjectures (theorems)
$\mathbb{Z}_p$, $\mathbb{Q}_p$	local points; $p$-adic integration; formal groups
$\mathbb{C}$	Betti comparison; Hodge structure on $H^*(X(\mathbb{C}))$

Finite fields. Smooth projective $X/\mathbb{F}_q$: Weil conjectures (proved by Dwork, Grothendieck, Deligne) describe $$ Z(X,t)=\exp\Bigl(\sum_{n\ge 1}#X(\mathbb{F}_{q^n})\frac{t^n}{n}\Bigr) $$ as rational, with functional equation and Riemann hypothesis $|\alpha|=q^{i/2}$ for eigenvalues of Frobenius on $H^i_{\mathrm{\acute{e}t}}$.

Rational points¶

$X(K)=\mathrm{Hom}_K(\mathrm{Spec} K,X)$ (for projective $X\subset\mathbb{P}^n$, $K$-points are projective coordinates in $K$ up to $K^\times$).

Geometry	Arithmetic expectation (rough)
Genus $0$ curve rational	$X(K)$ empty or dense (conics: Hasse)
Genus $1$	$X(K)$ torsor under $E(K)$; may fail Hasse
Genus $\ge 2$ (curve)	$X(K)$ finite (Faltings)
Abelian variety	$A(K)$ f.g.\ (Mordell–Weil)
Fano / rationally connected	expect many points after extension (conjectural packages)
General type	expect $X(K)$ not dense (Lang-type conjectures)

Weak approximation. $X(K)$ dense in $\prod_{v\in S}X(K_v)$ for finite $S$? Fails in presence of Brauer–Manin obstruction. Hardy–Littlewood / circle method treat complete intersections of low degree analytically — analytic NT.

Heights¶

On $\mathbb{P}^n(\overline{\mathbb{Q}})$, the absolute multiplicative height $$ H(x_0:\cdots:x_n)=\prod_v \max_i|x_i|_v $$ (normalized absolute values; product formula $\Rightarrow$ well-defined). Logarithmic $h=\log H$.

Northcott. $\{P\in\mathbb{P}^n(\overline{\mathbb{Q}}): H(P)\le B,\ [\mathbb{Q}(P):\mathbb{Q}]\le D\}$ is finite.

Canonical height $\hat{h}_L$ on abelian varieties for symmetric ample $L$ (Néron–Tate): quadratic form on $A(K)\otimes\mathbb{R}$, kernel $=$ torsion. Descent procedures use height machine to prove Mordell–Weil finiteness of rank generation.

Property	Statement
Functoriality	heights transform under morphisms with controlled error
Northcott	finite many bounded height + degree
Equidistribution	successive small-height points equidistribute (Szpiro–Ullmo–Zhang, …)

Faltings’ theorem (one-liner)¶

Theorem (Faltings, 1983; formerly Mordell conjecture). A smooth projective curve of genus $g\ge 2$ over a number field $K$ has only finitely many $K$-rational points.

Method (sketch). Finiteness of isogeny classes of abelian varieties over $K$ with good reduction outside a fixed set (via heights on moduli $\mathcal{A}_g$ and semisimplicity of Galois representations on Tate modules); then finiteness of $K$-points on subvarieties of abelian varieties (Mordellic) by specialization / Galois. Vojta / Bombieri later gave Diophantine approximation proofs for the curve case.

Corollaries. Fermat curve $x^n+y^n=z^n$ in $\mathbb{P}^2$ has genus $\ge 2$ for $n\ge 4$, hence finitely many rational points — but FLT needs modularity for the sharper “no nontrivial points.”

Étale cohomology pointer¶

For $X$ smooth separated over a field $k$, $\ell\neq\mathrm{char}\,k$, $$ H^i_{\mathrm{\acute{e}t}}(X_{\overline{k}},\mathbb{Q}_\ell) $$ is a finite-dim.\ $\mathbb{Q}_\ell$-vector space with continuous $\mathrm{Gal}(\overline{k}/k)$-action. Comparison: over $\mathbb{C}$, matches Betti cohomology after $\otimes\mathbb{C}$. Frobenius at good primes of a model over $\mathcal{O}_K$ acts with characteristic polynomial related to zeta factors of $X$.

Galois representations from geometry. $$ \rho_{X,\ell}:\mathrm{Gal}(\overline{K}/K)\to\mathrm{GL}\bigl(H^i_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_\ell)\bigr) $$ are the geometric input to Langlands and to modularity theorems (e.g.\ $i=1$, $X=E$ elliptic curve $\Rightarrow$ Tate module).

Computational tools. Lefschetz trace formula in étale cohomology computes $\#X(\mathbb{F}_q)$ from traces of Frobenius; spectral sequences as in spectral sequences and AG cohomology.

Local–global and descent (compact)¶

Selmer groups control $E(K)/nE(K)$ and $\mathrm{Sha}(E/K)[n]$ via Galois cohomology $H^1(K,E[n])$.
Descent on torsors cuts $X(K)$ into finer covering collections.
Brauer–Manin: $\mathrm{Br}(X)$ pairs with adelic points to obstruct rational points.

See it¶

Elliptic curves as the central testing ground. Mordell–Weil: $E(K)\cong E(K)_{\mathrm{tors}}\oplus\mathbb{Z}^r$. Rank $r$ is mysterious (BSD: $r=\mathrm{ord}_{s=1}L(E,s)$). Modular parametrization $X_0(N)\to E$ (modularity) identifies $L(E,s)$ with a modular $L$-function — arithmetic geometry + Langlands + analytic NT in one object.

Weil conjectures for curves. For smooth projective curve of genus $g$ over $\mathbb{F}_q$, $Z(X,t)=\frac{P(t)}{(1-t)(1-qt)}$ with $P$ degree $2g$, roots of absolute value $q^{-1/2}$ — RH for curves, proved by Weil; model for Deligne’s general proof.

Check¶

Outline why Northcott + canonical height imply $E(K)_{\mathrm{tors}}$ is finite for an elliptic curve over a number field.

Torsion points have $\hat{h}=0$. Difference $|\hat{h}-h|$ bounded on $E(\overline{K})$; so torsion has bounded naïve height. Northcott $\Rightarrow$ only finitely many algebraic points of bounded height and bounded degree — apply to $E(K)\subset E(\overline{K})$ (degree $\le[K:\mathbb{Q}]$).

How does good reduction of $E/\mathbb{Q}$ at $p$ constrain $\rho_{E,\ell}|_{I_p}$ for $\ell\neq p$?

Néron–Ogg–Shafarevich: good reduction $\Leftrightarrow$ $I_p$ acts trivially on $T_\ell(E)$. Hence the local Galois representation at $p$ is unramified for $\ell\neq p$, and $\mathrm{Frob}_p$ characteristic polynomial is $T^2-a_p T+p$ with $\#E(\mathbb{F}_p)=p+1-a_p$.

Pitfalls¶

Minimal model vs arbitrary integral model: reduction type (Kodaira symbols) is a property of the minimal regular model / Néron model.
Geometric genus vs arithmetic genus after bad reduction / singularities of models.
Faltings gives finiteness, not a bound algorithm in the original proof; effective Mordell is still hard in general.
Étale $\neq$ Zariski cohomology: higher étale cohomology of constructible sheaves is the arithmetic-rich theory; Zariski $H^i$ vanishes for $i>\dim$ quasi-coherent reasons differently.

Refs¶

Hartshorne, Algebraic Geometry (geometric baseline)
Silverman, AEC + Advanced Topics in the Arithmetic of Elliptic Curves
Hindry–Silverman, Diophantine Geometry
Milne, Étale Cohomology; Deligne, Weil I–II
Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern

Next¶

Arithmetic geometry index · Mordell–Weil & BSD · Diophantine & Arakelov · Étale & Weil · Galois Reps & Fontaine–Mazur · AG foundations · AG schemes · AG cohomology · Algebraic NT · Class field · Langlands · L-functions

Base	Typical questions
\(\mathbb{Q}\) or \(K\)	\(X(K)\) finite/infinite? density? Hasse principle?
\(\mathbb{F}_q\)	\(\#X(\mathbb{F}_{q^n})\); zeta function; Weil conjectures (theorems)
\(\mathbb{Z}_p\), \(\mathbb{Q}_p\)	local points; \(p\)-adic integration; formal groups
\(\mathbb{C}\)	Betti comparison; Hodge structure on \(H^*(X(\mathbb{C}))\)

Geometry	Arithmetic expectation (rough)
Genus \(0\) curve rational	\(X(K)\) empty or dense (conics: Hasse)
Genus \(1\)	\(X(K)\) torsor under \(E(K)\); may fail Hasse
Genus \(\ge 2\) (curve)	\(X(K)\) finite (Faltings)
Abelian variety	\(A(K)\) f.g.\ (Mordell–Weil)
Fano / rationally connected	expect many points after extension (conjectural packages)
General type	expect \(X(K)\) not dense (Lang-type conjectures)