Diophantine Geometry of Algebraic Curves

Diophantine geometry of algebraic curves explores the solutions of polynomial equations in integers or rational numbers by combining techniques from algebraic geometry, number theory and arithmetic analysis. Central to the field is the classification of curves by genus, with the celebrated Mordell–Faltings theorem guaranteeing that a curve of genus greater than one defined over a number field admits only finitely many rational points. The study further investigates explicit bounds on the number and distribution of these points, and seeks effective methods to determine them in concrete cases. Core approaches include descent procedures via Galois cohomology, which bound the size of Mordell–Weil groups of Jacobians; p-adic analytic methods such as the Chabauty–Coleman technique, which exploits the interplay between p-adic integration and the rank of the Jacobian; and its modern non-abelian and bilinear extensions under the umbrella of quadratic Chabauty or Chabauty–Kim, designed to remove rank restrictions. Computational realisations of these ideas have become increasingly sophisticated, yielding algorithms that determine integral or rational points on high-genus curves and hyperelliptic families. Beyond pure theory, these advances inform cryptographic protocols, influence the study of rational parametrisations and underscore deep connections between Diophantine finiteness, L-function special values and arithmetic statistics.

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Technical terms

Genus: A non-negative integer measuring the complexity of a curve, corresponding to the number of “holes” in its complex analytic form and controlling finiteness of rational points.

Jacobian: An abelian variety attached to a curve that parametrises degree-zero divisor classes, whose group of rational points encodes solutions to the original curve via the Abel–Jacobi map.

Mordell–Weil group: The finitely generated abelian group of rational points on a Jacobian over a number field, central to determining rational points on the curve itself.

Selmer group: A subgroup of Galois cohomology that bounds the Mordell–Weil group and plays a key role in descent methods for rank computations.

Chabauty–Coleman method: A p-adic technique that intersects the p-adic closure of the Mordell–Weil group in the Jacobian with the curve’s p-adic points to bound rational solutions when the rank is less than the genus.

Quadratic Chabauty: An extension of classical Chabauty that employs bilinear height pairings or non-abelian techniques to handle cases where the Mordell–Weil rank equals or exceeds the genus.

References

1. GEOMETRIC QUADRATIC CHABAUTY. Journal of the Institute of Mathematics of Jussieu (2021).
2. M0, 5: Toward the Chabauty–Kim method in higher dimensions. Mathematika (2023).
3. Quadratic Chabauty for modular curves: algorithms and examples. Compositio Mathematica (2023).
4. Computing torsion subgroups of Jacobians of hyperelliptic curves of genus 3. Research in Number Theory (2023).
5. Rational points on hyperelliptic Atkin-Lehner quotients of modular curves and their coverings. Research in Number Theory (2022).