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Seminars

Number Theory Seminar

Greatest Common Divisors and Integral Points on Projective Varieties

2026/11/06 (Fri.) 10:30~11:45
  • Date : 2026/11/06 (Fri.) 10:30~11:45
  • Location : Seminar Room 617, Institute of Mathematics (NTU Campus)
  • Speaker : Julie Tzu-Yueh Wang (Academia Sinica)
  • Abstract :
    Estimates for greatest common divisors, such as $\gcd(2^n-1,3^n-1)$, connect common divisibility with Diophantine approximation. In this talk, I will explain how these questions extend to integral points on projective varieties, where generalized GCD heights measure arithmetic proximity to subschemes of codimension at least two. Beginning with Silverman's interpretation of GCD heights via blowups and their connection with Vojta's conjecture, I will discuss joint work with Yasufuku that establishes unconditional estimates using the Ru--Vojta theorem. I will then present an alternative approach of Huang and Levin, based on Vojta's results on integral points on semi-abelian varieties. Finally, I will discuss recent joint work with Xiao extending the results of Huang, Levin, and Xiao from surfaces to higher-dimensional varieties. This work uses refined inequalities for vanishing along subschemes to obtain GCD estimates for intersections of boundary divisors.
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