Elliptic Arithmetic & Zeta Values

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Abstractions about elliptic units, periods, quadratic arithmetic, fundamental units, zeta functions, and number-theoretic criteria for elliptic curves.

7 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.

• Cube (algebra) — The third power x³ of a number or algebraic expression, obtained by multiplying three equal factors and inverted on suitable domains by the cube-root operation.
• Dedekind zeta function — Encode the nonzero ideals of a number field in a Dirichlet series and Euler product whose analytic behavior carries arithmetic information about the field.
• Elliptic unit — A distinguished algebraic unit in an abelian extension of an imaginary quadratic field, constructed from special values of modular or elliptic functions and forming an Euler system.
• Fundamental unit (number theory) — A generator, modulo roots of unity, of the rank-one unit group of a number field's ring of integers.
• Heegner number — One of the nine square-free positive integers d for which the imaginary quadratic field Q(√−d) has class number one, equivalently unique factorization in its ring of integers.
• Quarter period — The complete elliptic-integral quantities K(m) and iK′(m) that generate the period lattice of Jacobi elliptic functions and locate their characteristic quarter-cycle values.
• Tunnell's theorem — Test a square-free integer for the congruent-number property through equalities among counts of representations by four ternary quadratic forms—necessary unconditionally and sufficient conditional on Birch–Swinnerton-Dyer.