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| Lang (conjectures) | |
|---|---|
| Name | Serge Lang (conjectures) |
| Fields | Number theory, Algebraic geometry, Diophantine geometry |
Lang (conjectures)
Serge Lang formulated a suite of influential conjectures in number theory and algebraic geometry that relate Diophantine properties of varieties to their geometric and arithmetic structure. These conjectures connect problems studied by figures such as David Hilbert, Gérard Faltings, André Weil, Alexander Grothendieck, and John Tate and have inspired work by researchers including Pierre Deligne, Enrico Bombieri, Serge Lang, Joseph Silverman, and Andrew Wiles. The conjectures span topics touched by the Mordell conjecture, Birch and Swinnerton-Dyer conjecture, Manin conjecture, Mumford–Tate conjecture, and the Tate conjecture.
Lang proposed conjectures that broadly assert relationships among rational points, integral points, height functions, and the geometric type of algebraic varieties. His ideas interact with theorems and conjectures of Niels Henrik Abel, Carl Friedrich Gauss, Alexander Grothendieck, André Weil, and David Mumford, and are framed using tools from the work of John Milnor, Jean-Pierre Serre, Gerd Faltings and Paul Vojta. Lang's program influenced later results of Faltings, Bombieri, Enrico Bombieri and Walter Gubler, Joseph H. Silverman, and Shou-Wu Zhang.
Lang formulated conjectures predicting finiteness or sparsity of rational and integral points on varieties of certain types, echoing themes from the Mordell conjecture, the Shafarevich conjecture, and the Tate conjecture. He proposed that varieties of general type should have only finitely many rational points over number fields, a statement related to work by Gerd Faltings, Paul Vojta, Paul Gallagher, and Michael Hindry. Lang also conjectured analogues for integral points inspired by the Siegel theorem and linked to the ABC conjecture and conjectures studied by Enrico Bombieri, Alan Baker, and Yuri Manin.
Lang's geometric conjectures propose that positivity properties such as ampleness and canonical bundle behaviour control arithmetic distribution, invoking notions introduced by Alexander Grothendieck, Jean-Pierre Serre, David Mumford, and Phillip Griffiths. He connected the concept of varieties of general type to the arithmetic notions used in the Birch and Swinnerton-Dyer conjecture and the Mumford–Tate conjecture, drawing on foundational work by Grothendieck and later developments by Claire Voisin, Carlos Simpson, and Mihnea Popa. These conjectures often reference height functions developed by Shou-Wu Zhang, Joseph Silverman, and methods from Arakelov theory advanced by Gerd Faltings and Armand Borel.
Partial results validating Lang's expectations have been proved in special cases: the Mordell conjecture proven by Gerd Faltings settled finiteness for curves of genus greater than one over number fields; results by Paul Vojta and Enrico Bombieri provided conditional and unconditional progress on higher-dimensional analogues; and work by Faltings, Jean-Pierre Serre, Barry Mazur, and Andrew Wiles supplied tools impacting Lang's conjectures. Further progress includes results on rational points on subvarieties of abelian varieties by Faltings and achievements related to the Manin conjecture by Yuri Tschinkel and Tim Browning. Conditional implications have been shown using the ABC conjecture studied by Joseph Oesterlé and David Masser, and effective results derive from methods of Alan Baker and Enrico Bombieri.
Lang's conjectures link to broad swathes of arithmetic geometry and Diophantine approximation, connecting to the ABC conjecture, the Birch and Swinnerton-Dyer conjecture, the Mumford–Tate conjecture, and the Tate conjecture. They inform problems studied by David Mumford, John Tate, Alexander Grothendieck, Pierre Deligne, and Jean-Pierre Serre, and have consequences for the distribution of rational points on varieties considered by Yuri Manin and André Weil. The conjectures motivate techniques in Arakelov theory, heights, moduli spaces, and Hodge theory used by Claire Voisin, Carlos Simpson, Caucher Birkar, and Christopher Hacon.
Major open problems include proving Lang's predictions for higher-dimensional varieties of general type, establishing effective bounds for integral points in families as envisioned by Paul Vojta and confirming implications with the ABC conjecture explored by Yuri Bilu, Ken Ribet, and Shinichi Mochizuki. Active research engages mathematicians such as Akshay Venkatesh, Jacob Tsimerman, Aise Johan de Jong, Brian Conrad, and Keerthi Madapusi Pera on related questions about rational points, moduli of varieties, and relations to automorphic forms studied by Robert Langlands and Pierre Deligne. Progress may draw on advances in p-adic Hodge theory, derived algebraic geometry, and techniques from the proof of the Mordell conjecture and the Modularity theorem associated with Andrew Wiles and Richard Taylor.