20230718 
15:1516:00 
20230718,15:1516:00  LR1 (A31 3F) 
0718 Afternoon Math Lecture Room 1 (A31 3F)

Speaker 
The AxSchanuel Conjecture on Shimura varieties In the subject of number theory, the classical Lindemann theorem, which established that $e^{\alpha_1},\cdots,e^{\alpha_n}$ are algebraically independent whenever $\alpha_1,\cdots, \alpha_n$ are algebraic numbers linearly independent over the rationals, is a striking discovery in transcendence theory towards the end of the 19th century. As its extensive generalization, the Schanuel conjecture, according to which the field $\mathbb{Q}\left(\alpha_1,\cdots,\alpha_n; e^{\alpha_1},\cdots,e^{\alpha_n}\right)$ must be of transcendence degree $\geq n$ whenever $\alpha_1, \cdots, \alpha_n$ are $\mathbb{Q}$linearly independent complex numbers, has become a core problem of transcendental number theory. The AxSchanuel conjecture, which was resolved in the affirmative by Ax in 1971, was the analogue of the Schanuel conjecture on function fields dealing with the exponential function. The AxSchanuel conjecture on Shimura varieties is the analogue in which the exponential map $\mathrm{exp}: \mathbb{C}^n \to (\mathbb{C}^\ast)^n$ defined by $\mathrm{exp}(z_1,\cdots,z_n) := (e^{z_1},\cdots, e^{z_n})$ is replaced by the canonical projection map $\pi_\Gamma: \Omega \to \Omega/\Gamma =: X_\Gamma$ from a bounded symmetric domain $\Omega$ to a quotient Shimura variety $X_\Gamma$ corresponding to an arithmetic lattice $\Gamma \subset \mathrm{Aut}(\Omega)$. The theory of ominimal structures in mathematical logic especially the counting theorem of PilaWilkie (2006), coupled with complex differential geometry and monodromy results on Shimura varieties of Deligne, have led to results on functional transcendence theory notably the AxLindemann theorem of KlinglerUllmoYafaev (2016) and the AxSchanuel theorem for the $j$function of PilaTsimerman (2016). Coming from a completely different angle Mok (2019) introduced methods of complex geometry notably those on moduli schemes and their compactifications into functional transcendence theory for not necessarily arithmetic lattices. The latter perspective, coupled with the aforementioned methods and results, together with the theory of tame complex geometry of PeterzilStrachenko, has led to the proof of the AxSchanuel theorem for Shimura varieties by MokPilaTsimerman (2019). We will also discuss some farreaching applications of the theorem and its generalizations to number theory notably the uniform MordellLang theorem of DimitrovGaoHabegger (2021) on rational points.
