Mordell equation

Mordell equation
BradBeattie at English Wikipedia · CC BY-SA 3.0 · source
Name	Mordell equation
Field	Number theory
Introduced by	Louis Mordell
First published	1922
Equation	y^2 = x^3 + k
Variables	x, y, k
Notable solutions	Ramanujan, Mordell, Baker

Mordell equation is a family of Diophantine equations of the form y^2 = x^3 + k, where x, y are integers and k is a fixed nonzero integer constant. The equation lies at the intersection of arithmetic geometry, analytic number theory, and algebraic number theory, and connects with research of Louis Mordell, Srinivasa Ramanujan, Alexander Grothendieck, Gerd Faltings, and Alan Baker. It exemplifies deep links between elliptic curves, rational points, and Diophantine finiteness theorems studied at institutions such as University of Cambridge, Trinity College, Cambridge, and projects associated with the Clay Mathematics Institute.

Definition and statement

The equation studied is y^2 = x^3 + k for integer parameters x, y, and fixed integer k. For each fixed k the affine curve defined over Q is an elliptic curve after projective completion provided the cubic has distinct roots, linking to work at University of Manchester by Louis Mordell and later structural frameworks by André Weil and Jean-Pierre Serre. Integral solutions correspond to S-integral points on the corresponding elliptic curve, a viewpoint exploited by researchers at Princeton University and Institute for Advanced Study.

Historical context and Mordell's conjecture

Interest in specific values of k goes back to investigations by Srinivasa Ramanujan and examples communicated to G. H. Hardy and contemporaries in Cambridge. Louis Mordell formulated finiteness statements about rational points on curves and conjectured general finiteness for higher genus curves, leading to what became known as Mordell's conjecture. The conjecture was resolved by Gerd Faltings in the 1980s, building on techniques developed by Alexander Grothendieck in Paris seminars and subsequent work of Jean-Pierre Serre, John Tate, and Barry Mazur. The resolution had implications for specializations such as integral points on elliptic curves studied by Alan Baker, Enrico Bombieri, and the European Mathematical Society community.

Methods of solution and techniques

Techniques applied include descent methods introduced by Louis Mordell and formalized by S. M. R. K. Hodges and John Cremona, the theory of elliptic curves developed by Andrew Wiles and Barry Mazur, as well as transcendence methods of Alan Baker for linear forms in logarithms. Modular approaches connecting elliptic curves to modular forms, used prominently in the proof of the Taniyama–Shimura–Weil conjecture and by Jean-Pierre Serre and Ken Ribet, have influenced methods for solving particular instances. Computational algebra systems designed at University of Washington and algorithms of John Cremona and Nils Bruin enable practical searches for integer points using techniques from L-functions and Modular Forms Database collaborations.

Diophantine properties and finiteness results

For fixed nonzero k the set of integer solutions is finite except in special parameter families linked to complex multiplication by orders studied by G. H. Hardy and Emil Artin. Mordell’s original results about rational points established key finiteness properties later generalized by Faltings to higher genus. Results by Enrico Bombieri, Walter L. Baily Jr., and André Weil on height functions, and by Dorian Goldfeld on ranks of elliptic curves, control the behavior of rational and integral points. Baker’s effective bounds give explicit albeit large height bounds for solutions in many cases, a line of work continued by Alan Lauder and computational teams at University of Warwick.

Examples and computational results

Famous explicit examples include curves with k = 1, k = -1, and k = 2, which produced notable solutions found by Srinivasa Ramanujan and catalogued in tables compiled by John Cremona and contributors to the L-functions and Modular Forms Database. Computational searches by groups at University of Bristol and Max Planck Institute for Mathematics have enumerated solutions for many k within practical bounds, using algorithms influenced by Nils Bruin, Noam Elkies, and Michael Stoll. Techniques from computational algebraic geometry at Massachusetts Institute of Technology and numerical experiments recorded by Christopher Skinner and colleagues have further illustrated distributional patterns of integer solutions versus Mordell–Weil ranks studied by Richard Taylor.

Generalizations and related equations

Generalizations include Mordell-type equations y^2 = x^3 + ax + b (general elliptic curves) studied by André Weil and Serre-Tate theory, Thue–Mahler equations investigated by Kurt Mahler and Otto Thue, and superelliptic equations of form y^n = f(x) researched by Gerd Faltings and Philippon. Relations to the Birch and Swinnerton-Dyer conjecture link the arithmetic of the curve to analytic invariants studied by Bryan Birch, Peter Swinnerton-Dyer, and researchers at Harvard University and University of Cambridge. Modern extensions incorporate modularity lifting theorems of Richard Taylor and Andrew Wiles and effective methods from transcendence theory by Alan Baker and computational improvements by John Cremona.

Category:Diophantine equations