Diophantine geometry, a rather elegant term coined by Serge Lang, is where the formidable power of algebraic geometry is marshaled to dissect Diophantine equations. It’s a field that emerged as mathematicians in the 20th century realized these equations, often concerning integers, were far more tractable when viewed through a geometric lens. Essentially, Diophantine geometry is a sophisticated subset of the broader discipline of arithmetic geometry, which itself is a fascinating intersection of number theory and algebraic geometry.
Four pillars stand out in the architecture of Diophantine geometry, each a testament to profound insights: the Mordell–Weil theorem, Roth's theorem, Siegel's theorem, and Faltings's theorem. These aren't just abstract theorems; they represent crucial breakthroughs in understanding the structure and finiteness of solutions to certain types of equations.
Background
Serge Lang’s 1962 book, Diophantine Geometry, is credited with not only popularizing but also crystallizing this field under that very name. Before Lang, the study of Diophantine equations was often organized by the degree of the polynomial and the number of variables involved, a method reflected in Louis Mordell's own influential work, Diophantine Equations (1969). Mordell’s book, for instance, begins with a remark attributed to C. F. Gauss regarding homogeneous equations over the rational field. Gauss noted that if a non-zero solution exists in rational numbers, then a non-zero solution in integers (specifically, primitive lattice points) also exists. L. E. Dickson later added a nuance about parametric solutions, highlighting the complexities even in seemingly simple cases.
The historical development within this field is rich. The 1890 result by Hilbert and Adolf Hurwitz, which reduced the Diophantine geometry of curves of genus 0 to the study of degrees 1 and 2—essentially conic sections—finds a place in Mordell's book. Similarly, Siegel's theorem on integral points and Mordell's conjecture are discussed. Chapter 16 of Mordell's work delves into Mordell's theorem, which establishes the finite generation of the group of rational points on an elliptic curve. Chapter 26, meanwhile, tackles integer points on the specific case of the Mordell curve.
Mordell, in a rather pointed review of Lang's book, offered his perspective:
In recent times, powerful new geometric ideas and methods have been developed by means of which important new arithmetical theorems and related results have been found and proved and some of these are not easily proved otherwise. Further, there has been a tendency to clothe the old results, their extensions, and proofs in the new geometrical language. Sometimes, however, the full implications of results are best described in a geometrical setting. Lang has these aspects very much in mind in this book, and seems to miss no opportunity for geometric presentation. This accounts for his title "Diophantine Geometry."
Mordell observed that Lang's book largely focused on versions of the Mordell–Weil theorem, the Thue–Siegel–Roth theorem, and Siegel's theorem. Lang also included a treatment of Hilbert's irreducibility theorem and its applications, in the vein of Siegel's work. The significant divergence between their books, beyond stylistic differences, lay in Lang's use of abelian varieties and his presentation of a proof for Siegel's theorem, which Mordell described as being "of a very advanced character."
Despite an initial cool reception from some quarters, Lang's vision has proven remarkably influential. A 2006 tribute even lauded his book as "visionary." The field has expanded considerably, with the "arithmetic of abelian varieties" now encompassing Diophantine geometry alongside other sophisticated areas like class field theory, complex multiplication, local zeta-functions, and L-functions. As Paul Vojta noted, while figures like André Weil, John Tate, and Jean-Pierre Serre shared Lang's viewpoint, it’s easy to overlook that this perspective wasn't universally embraced at the time, as Mordell's review clearly demonstrates.
Approaches
The study of Diophantine geometry often begins with a single equation, which geometrically defines a hypersurface. When dealing with simultaneous Diophantine equations, we encounter a more general algebraic variety V over a field K. The central question then becomes the nature of the set V(K), comprising all points on V whose coordinates belong to K. Furthermore, by employing height functions, mathematicians can investigate quantitative aspects of these solutions—their "size"—as well as qualitative issues like existence and whether there are infinitely many.
Given the geometric underpinnings, the use of homogeneous equations and homogeneous co-ordinates is fundamental. This mirrors the prevalence of projective geometry in general algebraic geometry, as it allows for a unified treatment of points at infinity. Consequently, rational number solutions are the primary focus, though integer solutions—integer lattice points—can be handled by considering an affine variety embedded within a larger projective variety, thus incorporating those crucial points at infinity.
Faltings's theorem, which confirms a conjecture by L. J. Mordell, serves as a prime example of the general approach. It asserts that an algebraic curve C with genus g > 1 over the rational numbers possesses only a finite number of rational points. The earliest precursor to such a result is likely the theorem by Hilbert and Hurwitz for the case g = 0. The field is a dynamic landscape of established theorems, ongoing conjectures, and persistent open questions, each pushing the boundaries of our understanding.
The structure of geometry itself is vast, encompassing numerous branches. From the foundational Euclidean and Non-Euclidean geometries to more abstract realms like Algebraic, Arithmetic, Differential, and Topology, each offers a unique perspective on space, shape, and dimension. Diophantine geometry occupies a specific, yet crucial, niche within this landscape, bridging the gap between number theoretic problems and their geometric interpretations. It is a testament to the interconnectedness of mathematical disciplines, where the tools developed in one area can unlock profound insights in another. The pursuit of understanding solutions to equations, a task as old as mathematics itself, finds new and powerful expression in the sophisticated framework of Diophantine geometry.