Speaker : Julie Tzu-Yueh Wang (Institute of Mathematics, Academia Sinica)
Title : Integral points and analytic curves related to orbifolds
Time : 2021-09-24 (Fri) 13:30 - 14:45
Place : Seminar Room 617, Institute of Mathematics (NTU Campus)
In 1995, Henri Darmon and Andrew Granville proved: For any given positive integers $p, q, r$ satisfying $\frac 1p+\frac 1q+\frac 1r<1$, the generalized Fermat equation $x^p+y^q=z^r$ has only finitely many coprime integer solutions. Their proof is under the frame work of "curves with multiplicities" or "$M$-curves", which can be interpreted as orbifolds in the sense of Campana.

In this talk, we first introduce the notion of Campana's orbifold pairs and discuss some related results of Levin and Yasufuku on integral points. I will then mention the construction of a family of fibred threefolds $X_m \to (Y,\triangle)$ such that $X_m$ has no étale cover that dominates a variety of general type but it dominates the orbifold pairs $(Y,\triangle)$ of general type. Following Campana, the threefolds $X_m$ are called weakly special but not special. The Weak Specialness Conjecture predicts that a weakly special variety defined over a number field has a potential dense set of rational points. We prove that if $m$ is big enough, the threefolds $X_m$ present behaviors that contradict the analytic analogue of the Weak Specialness Conjecture. The second part of the talk is a joint work with Erwan Rousseau and Amos Turchet.

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