Plane curves

• Plane curve constructors
• Affine plane curves over a general ring
• Projective plane curves over a general ring
• Generic plane curves
• Plane conic constructor
• Projective plane conics over a field
• Projective plane conics over a number field
• Projective plane conics over \(\QQ\)
• Projective plane conics over finite fields
• Projective plane conics over prime finite fields
• Base class for Jacobians of curves

Plane quartics

• Quartic curve constructor
• Plane quartic curves over a general ring. These are generic genus 3 curves,

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

• Isomorphisms between Weierstrass models of elliptic curves
• Isogenies
• Isogenies of small prime degree.

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
• Torsion subgroups of elliptic curves over number fields (including \(\QQ\))
• 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

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
• Miscellaneous \(p\)-adic functions

Analytic properties over \(\CC\).

• Weierstrass \(\wp\)-function for elliptic curves
• Period lattices of elliptic curves and related functions
• Regions in fundamental domains of period lattices

Modularity and \(L\)-series over \(\QQ\).

• Modular parametrization of elliptic curves over \(\QQ\)
• Modular symbols
• L-series for elliptic curves
• Heegner points on elliptic curves over the rational numbers
• p-adic L-functions of elliptic curves

To be sorted

• Descent on elliptic curves over \(\QQ\) with a 2-isogeny.
• Elliptic curves with prescribed good reduction.
• Elliptic curves over padic fields
• Denis Simon’s PARI scripts
• 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 padic 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 over GF(p),
• 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
• Hyperelliptic curves of genus 2 over a general ring
• Hyperelliptic curves of genus 2 over a finite field
• Hyperelliptic curves of genus 2 over a p-adic field
• Hyperelliptic curves of genus 2 over the rationals
• 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