Home /  Degeneracy of Algebraic Points

Workshop

Degeneracy of Algebraic Points April 24, 2023 - April 28, 2023
Registration Deadline: April 28, 2023 about 1 year ago
To apply for Funding you must register by: January 17, 2023 over 1 year ago
Parent Program:
Location: SLMath: Eisenbud Auditorium
Organizers Jennifer Balakrishnan (Boston University), LEAD Mirela Ciperiani (University of Texas, Austin), Philipp Habegger (University of Basel), Wei Ho (Institute for Advanced Study), Hector Pasten (Pontificia Universidad Católica de Chile), Yunqing Tang (University of California, Berkeley), Shou-Wu Zhang (Princeton University)
Speaker(s)

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Description
Image
A genus 2 curve over the reals and various p-adics. Image created by Prof. Jennifer Balakrishnan .
A central topic in Diophantine Geometry is to understand how the geometry of a variety influences the arithmetic of its algebraic points, and conversely. Conjectures of Bombieri, Lang, and Vojta suggest that rational points of algebraic varieties satisfying suitable approximation conditions, are algebraically degenerate. On the other hand, conjectures on unlikely intersections suggest that algebraic points of special type —e.g. torsion points in semi-abelian varieties, special points in Shimura varieties— avoid subvarieties, unless the subvariety itself is also special (in a technical sense). In recent years, a number of techniques have led to outstanding progress on Lang-Vojta conjectures, such as the Subspace Theorem, p-adic approaches to finiteness, and modular methods. Similarly, spectacular progress has been achieved on unlikely intersection conjectures thanks to new methods and tools, such as height formulas for special points, connections to model theory, refined counting results, and new theorems of Ax-Shanuel type (bi-algebraic geometry). The goal of this workshop is to create the opportunity for these two groups to interact, to share their techniques, to update on the most recent progress, and to attack the outstanding open questions in the field. The two directions described above are rather technical and specialized, and it seems necessary to bring together these groups of researchers to explain to each other not only the latest developments in their fields, but also the methods that made possible these breakthroughs. Thus, in this workshop we expect to have lectures explaining the main methods, as well as talks presenting the most recent progress in the subject by the world leading experts.
Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • Arithmetic geometry

  • Arakelov theory

  • unlikely intersections

  • rational points

  • integral points

  • CM points

  • Colmez’s conjecture

  • CM abelian varieties

  • Vojta’s conjecture

  • Grothendieck-Katz p-curvature conjecture

  • Faltings height

  • Chabauty’s method

  • non-abelian Chabauty

  • period mappings

  • Bombieri-Lang conjecture

  • Lang’s conjecture

  • Effectivity

  • Diophantine equations

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification
Schedule, Notes/Handouts & Videos
Show Schedule, Notes/Handouts & Videos
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Apr 24, 2023
Monday
09:15 AM - 09:30 AM
  Welcome
09:30 AM - 10:30 AM
  On Effectivity in Some Diophantine Problems
Umberto Zannier (Scuola Normale Superiore)
10:30 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Integer Matrices with a Given Characteristic Polynomial and Multiplicative Dependence of Matrices
Alina Ostafe (University of New South Wales)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Effective Counting and Families of Abelian Varieties
Harry Schmidt (Universität Basel)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Modularity and Effective Mordell
Levent Alpöge (Harvard University)
Apr 25, 2023
Tuesday
09:30 AM - 10:30 AM
  Bigness of the Admissible Canonical Bundle
Xinyi Yuan (Peking University)
10:30 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Preperiodic Points in Families of Rational Maps
Niki Myrto Mavraki (Harvard University)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Reduction of Brauer Classes on K3 Surfaces
Salim Tayou (Harvard University)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Computing the Weil Representation of a Superelliptic Curve
Irene Bouw (Ulm University)
04:30 PM - 06:20 PM
  Reception
Apr 26, 2023
Wednesday
09:30 AM - 10:30 AM
  Heights in the Isogeny Class of an Abelian Variety
Mark Kisin (Harvard University)
10:30 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Crystallinity Properties of Complex Rigid Local Systems (Joint work in progress with Michael Groechenig)
Hélène Esnault (Freie Universität Berlin)
12:00 PM - 02:00 PM
  Lunch
03:00 PM - 03:30 PM
  Afternoon Tea
Apr 27, 2023
Thursday
09:30 AM - 10:30 AM
  Degeneracy Loci in Families of Abelian Varieties and their Applications
Ziyang Gao (Leibniz Universität Hannover)
10:30 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  A Twisting-Free Converse Theorem for GL(2)
Vesselin Dimitrov (Institute for Advanced Study)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Some New Elliptic Integrals
David Masser (Universität Basel)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  A P-Adic Analogue of Borel's Theorem
Ananth Shankar (University of Wisconsin-Madison)
Apr 28, 2023
Friday
09:30 AM - 10:30 AM
  Conditional Algorithmic Mordell
Brian Lawrence (University of Wisconsin-Madison)
10:30 AM - 11:00 AM
  Break
11:00 AM - 12:00 PM
  Joint Unlikely Almost Intersections on Ordinary Siegel Spaces
Congling Qiu (Yale University)
12:00 PM - 02:00 PM
  Lunch
02:00 PM - 03:00 PM
  Roth's Theorem Over Adelic Curves
Paul Vojta (University of California, Berkeley)
03:00 PM - 03:30 PM
  Afternoon Tea
03:30 PM - 04:30 PM
  Diophantine Geometry: All Our Yesterdays
Shou-Wu Zhang (Princeton University)