Arithmetic Geometry¶
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13 domain-specific abstractions whose origin domain is Arithmetic Geometry.
- Arakelov theory — An arithmetic geometry that augments schemes over the integers with analytic data at infinite places.
- Arithmetic surface — A regular or suitably controlled two-dimensional scheme fibered over the spectrum of a Dedekind domain, with generic fiber an algebraic curve.
- Brandt matrix — A matrix encoding counts or weighted correspondences among ideal classes of a definite quaternion algebra, realizing Hecke operators on quaternionic modular forms.
- F-crystal — A finite free Witt-vector module equipped with an injective Frobenius-semilinear endomorphism, encoding crystalline-cohomological Frobenius structure.
- Frobenioid — A category equipped with degree, divisor-like, and Frobenius structure that categorifies monoid actions arising from arithmetic line bundles.
- Heegner's lemma — A descent lemma stating that a quartic curve with nonsquare leading coefficient has a rational point if it has a point over an odd-degree extension.
- Higher local field — A field equipped with an iterated tower of complete discrete valuations whose final residue field is finite or otherwise specified at dimension zero.
- Hodge–Arakelov theory — A proposed Arakelov-geometric analogue of Hodge comparison theory for elliptic curves, centered on functions on universal extensions and torsion points.
- Mordell–Weil Rank of an Elliptic Curve — Count the independent infinite-order rational points on an elliptic curve over a declared number field by taking the free rank of its finitely generated Mordell–Weil group.
- Mordell–Weil Theorem — For an abelian variety over a number field, the group of rational points is finitely generated.
- Néron–Tate height — A canonical quadratic height on rational points of an abelian variety over a global field.
- P-adic Hodge theory — A comparison theory classifying p-adic Galois representations through period rings and their relations to de Rham, crystalline, semistable and Hodge–Tate cohomology.
- P-derivation — A prime-indexed arithmetic analogue of derivation whose addition and product laws encode a lift of Frobenius modulo p.