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
- Fractional morphisms 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 \(\QQ\))
- 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
- Testing whether elliptic curves over number fields are \(\QQ\)-curves
The following relate to elliptic curves over local nonarchimedean fields.
Analytic properties over \(\CC\).
Modularity and \(L\)-series over \(\QQ\).
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 \(\QQ\) 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¶
- Constructor for hyperelliptic curves using the smooth model
- Hyperelliptic curves (smooth model) over a field
- Hyperelliptic curves (smooth model) over a finite field
- Hyperelliptic curves (smooth model) over a p-adic field
- Hyperelliptic curves (smooth model) over the rationals
- Hyperelliptic curves of genus 2 in the smooth model
- Compute invariants of quintics and sextics via ‘Ueberschiebung’
- Mestre’s algorithm
- Computation of Frobenius matrix on Monsky-Washnitzer cohomology
- Frobenius on Monsky-Washnitzer cohomology of a hyperelliptic curve
- Conductor and reduction types for genus 2 curves
Jacobians of hyperelliptic curves¶
- Jacobian of a general hyperelliptic curve
- Rational point sets on a Jacobian of a general hyperelliptic curve
- Rational point sets on a Jacobian of a hyperelliptic curve (ramified case)
- Rational point sets on a Jacobian of a hyperelliptic curve (split case)
- Rational point sets on a Jacobian of a hyperelliptic curve (inert case)
- Jacobians of genus-2 hyperelliptic curves
- Rational point sets of Jacobians of genus-2 curves (ramified case)
- Rational point sets of Jacobians of genus-2 curves (split case)
- Rational point sets of Jacobians of genus-2 curves (inert case)
- Arithmetic on the Jacobian