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Diophantine Equations: Special Points, Integrality, and Beyond

Research Project
 | 
01.04.2019
 - 31.03.2023

Describing the rational and integral solutions of diophantineequations is among the oldest problems in mathematics. In the 20thcentury great progress was made by Thue, Siegel, Faltings, Vojta, andmany others towards our understanding of rational and integral pointsin abelian varieties. Landmark results were conjectures of Mordell andLang. Special values of classical transcendental functions, such asthe exponential function, Weierstrass functions, and modular forms,are a rich source of points of arithmetic significance. Thesespecial points include roots of unity, points of finite orderon abelian varieties, and special points on Shimura varieties. TheConjectures of Manin-Mumford and André-Oort, now known to be true inmany important cases, underscore a striking similarity between specialpoints and rational points. Recent work also suggests a deepconnection between special points and integral points. This connectionis less well understood and lies outside the scope of theaforementioned conjectures. One purpose of this proposal is toinvestigate special points through the lens of Lang's Conjecture onintegral points. Furthermore, there are intriguing connections toergodic theory and dynamical systems. Another focus are the morerecent, and largely open, conjectures on unlikely intersections. Theyare due independently to Bombieri--Masser--Zannier, Pink, and Zilberand go the beyond the conjectures above in a different way. They havebeen the source of many positive interactions between number theoryand model theory.

Funding

Diophantine Equations: Special Points, Integrality, and Beyond

SNF Projekt (GrantsTool), 10.2019-09.2023 (48)
PI : Habegger, Philipp.

Members (4)

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Philipp Habegger

Principal Investigator
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Marta Dujella

Project Member
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Gerold Schefer

Project Member
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Robert Wilms

Project Member