Diophantine Geometry, Faltings, and Arakelov Theory¶
Job: develop the Faltings theorem (Mordell) / the Vojta / the Diophantine approximation, the height machine / Northcott / equidistribution / Arakelov intersection, the Nevanlinna / Vojta dictionary / abc, and the Lang / Mordell–Lang / Manin–Mumford / Bogomolov — with the Faltings via the Vojta / the Roth as the worked example.
Core¶
The Faltings theorem and the proof¶
Theorem (Faltings, 1983). The Mordell conjecture: the curve \(C/K\) genus \(g\ge2\) the \(|C(K)|<\infty\).
The proof (the Faltings 1983: the finiteness of the abelian varieties isogeny classes / the semisimplicity of the Galois / the Mordellic, the Vojta / the Bombieri the Diophantine approximation).
| Faltings / proof | Description |
|---|---|
| Faltings (Mordell) | the $ |
| Proof (Faltings) | the finiteness of the abelian varieties isogeny classes / the semisimplicity / the Mordellic |
| Vojta / Bombieri | the Diophantine approximation: the height inequality / the Roth theorem the arithmetic analog |
| Mordellic | the finiteness of the \(K\)-points on the subvarieties of the abelian varieties (the Mordellic) |
The height machine and the Northcott / the equidistribution¶
| Height machine / Northcott / Arakelov | Description |
|---|---|
| Height machine | \(h_L(X)=\deg(L)+O(1)\) (the functorial heights, the Hilbert / the Weil height) |
| Northcott | the finite / the bounded (the bounded height + degree the finite) |
| Equidistribution (Szpiro–Ullmo–Zhang) | the small points the equidistribution (the Galois / the height) |
| Arakelov / arithmetic intersection | the arithmetic variety \(\overline{\mathcal{X}}\) the fibration over \(\mathrm{Spec}\,\mathbb{Z}\), the Arakelov intersection / the Green's function, the height the arithmetic intersection |
The Nevanlinna / the Vojta dictionary and the abc conjecture¶
| Nevanlinna / Vojta / abc | Description |
|---|---|
| Nevanlinna theory | the holomorphic curves / the Cartan, the Second Main Theorem, the height / the characteristic |
| Vojta dictionary | the arithmetic / the Nevanlinna the analog (the Diophantine approximation / the holomorphic curves, the Roth / the Cartan) |
| abc conjecture (Masser–Oesterlé) | $\max( |
| Belyi / Mochizuki | the Belyi / the Mochizuki the approach (not the consensus) |
The Lang conjectures and the general type¶
| Lang / Mordell–Lang / Manin–Mumford | Description |
|---|---|
| Lang conjecture | the general type \(X\) the \(X(K)\) not dense (the Lang / the Bombieri–Lang) |
| Mordell–Lang | the Faltings / the Hrushovski: the subvariety of an abelian variety / the intersection with the finitely-generated subgroup |
| Manin–Mumford / Bogomolov | the torsion / the small points (the Raynaud / the Ullmo–Zhang) |
See it¶
Faltings theorem via the Vojta / the Diophantine approximation and the Roth theorem. The Roth theorem (the \(\alpha\) algebraic, the \(|\alpha-p/q|>c/q^{2+\varepsilon}\) for all but finitely many \(p/q\), the Diophantine approximation). The Vojta / the Faltings: the curve \(C\) genus \(g\ge2\), the heights / the canonical / the Arakelov, the Vojta the height inequality (the \(h(P)\le\varepsilon h(C)+O(1)\) the analog of the Roth / the Diophantine approximation), the Faltings the finiteness (the height inequality \(\Rightarrow\) the finite \(C(K)\)). The Arakelov: the arithmetic intersection / the Green's function the analytic height, the Vojta the height inequality the Arakelov-theoretic. The key point: the Faltings (the Mordell) via the Vojta / the Diophantine approximation / the Arakelov — the height machine the Roth / the Vojta, the arithmetic intersection the analytic / the geometric.
Check¶
Show that the Roth theorem is the arithmetic analog of the Nevanlinna Second Main Theorem, and explain the Vojta dictionary.
The Roth theorem: for \(\alpha\) algebraic irrational, \(|\alpha-p/q|>c(\alpha,\varepsilon)/q^{2+\varepsilon}\) for all but finitely many \(p/q\) (the Diophantine approximation, the exponent \(2+\varepsilon\) the Roth, the optimal the \(2\) the Dirichlet). The Nevanlinna Second Main Theorem: for a holomorphic curve \(f:\mathbb{C}\to X\) into a variety \(X\) (the Nevanlinna), the height (the characteristic \(T(r,f)\)) the Second Main Theorem (the height \(\le\) the ramification / the defect, the \(N_1(r,f)\) the error term). The Vojta dictionary: the arithmetic / the Nevanlinna the analog — the Diophantine approximation / the holomorphic curves (the \(p/q\to\alpha\) the \(f(z)\to X\)), the Roth / the Cartan (the Second Main Theorem), the height inequality / the Nevanlinna. The Vojta the dictionary (the analogy, not the theorem — the two the related, the arithmetic the Diophantine, the Nevanlinna the holomorphic).
Explain the Arakelov arithmetic intersection and its relation to the height, and identify the Green's function.
The Arakelov theory: the arithmetic variety \(\overline{\mathcal{X}}\) (the fibration over \(\mathrm{Spec}\,\mathbb{Z}\), the \(\mathcal{X}\to\mathrm{Spec}\,\mathbb{Z}\) the model, the Arakelov the compactification at the archimedean place). The arithmetic intersection: for divisors \(D_1,D_2\) on \(\overline{\mathcal{X}}\), the \(D_1\cdot D_2\) the Arakelov intersection (the finite part the intersection on the fibers + the archimedean the Green's function). The Green's function \(g(D)\) (the archimedean / the complex analytic, the logarithmic potential, the \(\int\log|z-w|d\mu\)). The height: \(h(P)=\overline{D}\cdot\overline{\mathcal{P}}\) (the arithmetic intersection of the divisor with the section, the Arakelov the height). So the height the Arakelov intersection (the arithmetic intersection the analytic / the Green's function the archimedean, the finite the fiber, the height the sum).
Pitfalls¶
- The Faltings the finiteness not the effective (the effective Mordell the open, the height bounds the open). The Faltings gives the finiteness, not the effective bound.
- The Vojta / the Nevanlinna the dictionary (the analogy, not the theorem — the abc the conjecture, the Belyi / the Mochizuki the approach not the consensus). The Vojta the dictionary the structural / the analogy, the abc the conjecture.
- The Lang / the general type the conjecture (the Bombieri–Lang the conjectural, not the proven). The Lang conjecture the general type \(X(K)\) not dense — the conjectural, not the proven.
- The Arakelov / the arithmetic intersection the analytic (the Green's function the archimedean), not the purely algebraic — the Arakelov the analytic / the arithmetic, the Green's function the transcendental.
- The Mordell–Lang / the Manin–Mumford the Faltings / the Hrushovski (the Mordell–Lang), the Raynaud / the Ullmo–Zhang (the Manin–Mumford / Bogomolov) — the two related but distinct (the Mordell–Lang the finitely-generated, the Manin–Mumford the torsion).
Refs¶
- Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" (Invent. 1983); Vojta, "A new proof of the Mordell conjecture" (1991)
- Bombieri, "The Mordell conjecture revisited" (Annali 1990); the Vojta / the Diophantine approximation
- Arakelov, "Intersection theory of divisors" (1974); Soulé, Lectures on Arakelov Geometry; the Green's function / the arithmetic intersection
- Lang, Number Theory III (the Lang conjectures); Raynaud, "Manin–Mumford" (1983); Ullmo–Zhang, "Equidistribution of small points" (Duke 1998)
Next¶
AG index · Foundations · Mordell–Weil & BSD · Étale & Weil · Galois Reps & Fontaine–Mazur · Algebraic geometry · Algebraic NT