Arithmetic Geometry & P-Adic Theory

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Abstractions about arithmetic surfaces, heights, local fields, p-adic Hodge theory, crystals, matrices, and interactions between number theory and geometry.

9 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• 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.
• 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.
• 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.