Elliptic curves

• Elliptic curve constructor
• Construct elliptic curves as Jacobians
• Points on elliptic curves
• Elliptic curves over a general ring
• Elliptic curves over a general field
• Elliptic curves over finite fields
• Formal groups of elliptic curves

Maps between them

• Elliptic-curve morphisms
• Composite morphisms of elliptic curves
• Sums of morphisms of elliptic curves
• Isomorphisms between Weierstrass models of elliptic curves
• Isogenies
• Square‑root Vélu algorithm for elliptic-curve isogenies
• Scalar-multiplication morphisms of elliptic curves
• Frobenius isogenies of elliptic curves
• Isogenies of small prime degree
• Modular polynomials for elliptic curves

Elliptic curves over number fields

• Elliptic curves over the rational numbers
• Tables of elliptic curves of given rank
• Elliptic curves over number fields
• Canonical heights for elliptic curves over number fields
• Saturation of Mordell-Weil groups of elliptic curves over number fields
• Torsion subgroups of elliptic curves over number fields (including $\mathbf{Q}$)
• Galois representations attached to elliptic curves
• Galois representations for elliptic curves over number fields
• Isogeny class of elliptic curves over number fields
• Tate-Shafarevich group
• Complex multiplication for elliptic curves
• Testing whether elliptic curves over number fields are $\mathbf{Q}$-curves

The following relate to elliptic curves over local nonarchimedean fields.

• Local data for elliptic curves over number fields
• Kodaira symbols
• Tate’s parametrisation of $p$-adic curves with multiplicative reduction
• Elliptic curves over $p$-adic fields

Analytic properties over $\mathbf{C}$.

• Weierstrass $\mathrm{\wp }$-function for elliptic curves
• Period lattices of elliptic curves and related functions
• Regions in fundamental domains of period lattices

Modularity and $L$-series over $\mathbf{Q}$.

• Modular parametrization of elliptic curves over $\mathbf{Q}$
• Modular symbols attached to elliptic curves over $\mathbf{Q}$
• Modular symbols by numerical integration
• $L$-series for elliptic curves
• Heegner points on elliptic curves over the rational numbers
• $p$-adic $L$-functions of elliptic curves

To be sorted

• Birch and Swinnerton-Dyer formulas
• Specific algorithms to compute cardinality of elliptic curves over a finite field
• Descent on elliptic curves over $\mathbf{Q}$ with a 2-isogeny
• Elliptic curves with prescribed good reduction
• Denis Simon’s PARI scripts
• Global and semi-global minimal models for elliptic curves over number fields
• Elliptic curves with congruent mod-5 representation
• Morphism to bring a genus-one curve into Weierstrass form

Hyperelliptic curves

• Hyperelliptic curve constructor
• Hyperelliptic curves over a general ring
• Hyperelliptic curves over a finite field
• Hyperelliptic curves over a $p$-adic field
• Hyperelliptic curves over the rationals
• Mestre’s algorithm
• Computation of Frobenius matrix on Monsky-Washnitzer cohomology
• Frobenius on Monsky-Washnitzer cohomology of a hyperelliptic curve
• Jacobian of a general hyperelliptic curve
• Jacobian of a hyperelliptic curve of genus 2
• Rational point sets on a Jacobian
• Jacobian ‘morphism’ as a class in the Picard group
• Some functions regarding geometric endomorphism rings of Jacobians of
• Hyperelliptic curves of genus 2 over a general ring
• Compute invariants of quintics and sextics via ‘Ueberschiebung’
• Kummer surfaces over a general ring
• Conductor and reduction types for genus 2 curves

Indices and Tables

• Index

• Module Index

• Search Page