Arithmetic Geometry¶
Job: study algebraic varieties over number fields and finite fields — rational points, heights, Galois representations on étale cohomology — where geometry meets arithmetic.
Core¶
Arithmetic geometry treats schemes / varieties defined over \(\mathbb{Z}\), \(\mathcal{O}_K\), \(\mathbb{F}_q\), or \(\mathbb{Q}\), emphasizing integral models, reduction mod \(\mathfrak{p}\), and Diophantine sets \(X(K)\). Tools: heights, Galois cohomology, étale cohomology, and modularity / Langlands when Galois representations appear.
| Layer | Content |
|---|---|
| Base fields | number fields, local fields, \(\mathbb{F}_q\) |
| Points | \(X(K)\), weak/strong approximation |
| Heights | Northcott; Faltings (one-liner) |
| Cohomology | étale \(H^i_{\mathrm{\acute{e}t}}\), Frobenius eigenvalues |
| Reciprocity | Galois reps \(\rightsquigarrow\) automorphic (Langlands) |
Reading map¶
| Page | Focus |
|---|---|
| Foundations | varieties over global/finite fields; points; heights; Faltings; étale pointer |
| Mordell–Weil & BSD | Mordell–Weil / canonical height / descent / Selmer; BSD / refined / Tamagawa / Sha; Gross–Zagier / Kolyvagin; Hasse–Weil \(L\) / modularity / \(p\)-adic BSD |
| Diophantine & Arakelov | Faltings / Vojta / Diophantine approximation; height machine / Northcott / equidistribution / Arakelov; Nevanlinna / Vojta / abc; Lang / Mordell–Lang / Manin–Mumford |
| Étale & Weil | étale / \(\ell\)-adic cohomology / comparison / finiteness; Weil / Dwork–Grothendieck–Deligne; Lefschetz trace / point counting; Deligne RH / weights / Weil II; crystalline / \(p\)-adic / prismatic |
| Galois Reps & Fontaine–Mazur | Galois reps from geometry / Tate module; Fontaine–Mazur / geometric; modularity / Langlands / Serre; \(p\)-adic Hodge / Fontaine / prismatic; Sha / Selmer / Cassels–Tate |
Dependency order¶
- Algebraic geometry (schemes, cohomology).
- Algebraic number theory.
- Galois theory.
- Rational points / heights \(\to\) étale cohomology \(\to\) Langlands.
Interfaces¶
| Area | Contact |
|---|---|
| Algebraic geometry | geometric methods over non-alg.\ closed fields |
| Algebraic NT | \(\mathcal{O}_K\)-models; class group as \(\mathrm{Pic}\) |
| Analytic NT | \(L\)-functions of varieties |
| Langlands | modularity of Galois reps |
| Homological algebra | Galois cohomology \(\mathrm{Ext}_{\mathbb{Z}[G]}\) |
| Derived AG | derived moduli of bundles / shtukas |
Notation (paper-standard)¶
- \(X/K\) variety; \(X(\overline{K})\), \(X(K)\); reduction \(X_{\mathbb{F}_p}\)
- \(H^i_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Q}_\ell)\); \(\mathrm{Frob}_p\)
- height \(H(P)\); canonical \(\hat{h}\) on abelian varieties
- \(\mathrm{Sha}\), Selmer groups (pointer)
See it¶
Hasse principle failure. \(X:\, 3x^3+4y^3+5z^3=0\) in \(\mathbb{P}^2\) is a famous Selmer curve with points over all \(\mathbb{Q}_v\) but not over \(\mathbb{Q}\). Local-global principles need Brauer–Manin / descent obstructions beyond naive Hasse — arithmetic geometry of rational points in one example.
Check¶
Why does the Hasse principle hold for quadrics but not for all genus-\(1\) curves?
Quadratic forms: Hasse–Minkowski theorem (local-global for \(\mathbb{Q}\)). Genus-\(1\): elements of \(\mathrm{Sha}(E/K)\) give torsors under elliptic curves that are locally trivial everywhere but globally nontrivial — precisely Hasse failures.
What does ‘reduction mod \(p\) is good’ mean for an abelian variety \(A/\mathbb{Q}\)?
Existence of an abelian scheme model over \(\mathbb{Z}_p\) (Néron model has good reduction) \(\Leftrightarrow\) inertia at \(p\) acts trivially on \(T_\ell(A)\) (Néron–Ogg–Shafarevich).
Pitfalls¶
- Geometric vs arithmetic irreducible fiber: reduction type depends on the integral model.
- Rational points vs algebraic points: \(X(\overline{K})\) huge; \(X(K)\) may be finite (Faltings).
- \(\ell\)-adic vs crystalline vs de Rham comparison theorems are subtle at \(p=\ell\).
Refs¶
- Silverman, The Arithmetic of Elliptic Curves; Advanced Topics
- Hindry–Silverman, Diophantine Geometry
- Milne, Étale Cohomology; Freitag–Kiehl
- Cornell–Silverman–Stevens, Modular Forms and Fermat’s Last Theorem
Next¶
Foundations · Mordell–Weil & BSD · Diophantine & Arakelov · Étale & Weil · Galois Reps & Fontaine–Mazur · Algebraic geometry · Algebraic NT · Langlands · Analytic NT · AG cohomology