Arithmetic Geometry of Elliptic Curves

Arithmetic geometry of elliptic curves combines algebraic geometry and number theory to investigate cubic curves of genus one endowed with a distinguished rational point. Such curves admit a group law under which their rational solutions form a finitely generated abelian group known as the Mordell–Weil group. The structure of this group—its rank and finite torsion subgroup—reflects deep Diophantine properties and is intertwined with the behaviour of associated L-functions. Isogenies, which are non-constant morphisms preserving the identity element, organise elliptic curves into isogeny classes and underpin modern modularity results. The proof of the modularity theorem for curves over the rationals established a correspondence between elliptic curves and modular forms, yielding explicit methods for point counting, the study of rational points and applications to cryptography. Central conjectures, most notably the Birch and Swinnerton-Dyer conjecture, predict a precise relation between the order of vanishing of the L-function at its central point and the rank of the Mordell–Weil group. Recent theoretical advances have been complemented by computational techniques that construct curves with specified torsion over finite fields, classify torsion over number fields of small degree and provide global criteria for congruences of Galois representations. These developments highlight a rich dialogue between abstract theory and explicit algorithmic realisations, with far-reaching implications for Diophantine equations, secure communications and the theory of automorphic forms.

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Technical terms

Elliptic curve: A smooth projective cubic curve of genus one, equipped with a distinguished rational point that serves as the identity for a natural group law.

Rational point: A solution to the defining equation of a curve whose coordinates lie in the field of rational numbers.

Mordell–Weil group: The abelian group formed by the rational points on an elliptic curve under the chord–tangent addition law.

Isogeny: A non-constant morphism between elliptic curves that preserves the designated identity point and group structure.

Torsion subgroup: The subset of rational points on an elliptic curve that have finite order under the group law.

Galois representation: A homomorphism from the absolute Galois group of a field into the automorphism group of a vector space, encoding arithmetic information of an elliptic curve.

L-function: A complex analytic function attached to an elliptic curve, formed as an Euler product over primes and reflecting its arithmetic and geometric properties.

References

1. Torsion points on elliptic curves over number fields of small degree. Algebra & Number Theory (2023).
2. Global methods for the symplectic type of congruences between elliptic curves. Revista Matemática Iberoamericana (2021).
3. Constructing elliptic curves over finite fields with prescribed torsion. Mathematics of Computation (2011).