The Hodge index theorem of Faltings and Hriljac asserts that the Neron-Tate height pairing on a projective curve over a number field is equal to a certain intersection pairing in the setting of Arakelov geometry. In the talk, I will present an extension of this result to adelic line bundles on higher dimensional varieties over finitely generated fields. Then I will talk about its relation to the non-archimedean Calabi-Yau theorem and its application to algebraic dynamics. This is a joint work with Shou-Wu Zhang.

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A classical result of Kronecker and Weber states that the value of the elliptic j-function at a point of complex multiplication (i.e. a point lying in the intersection of the upper half-plain and some imaginary quadratic field) is algebraic. B. Gross and D. Zagier have conjectured that a similar phenomenon also holds for certain modular eigenfunctions of the hyperbolic Laplace operator. Namely, the higher Green's functions are real-valued functions of two variables on the upper half-plane which are bi-invariant under the action of SL2(Z), have a logarithmic singularity along the diagonal and are eigenfunctions of the hyperbolic Laplace operator with eigenvalue k(1-k) for some positive integer k. The conjecture formulated in "Heegner points and derivatives of L-series'' (1986) predicts when the value of a higher Green's function at a pair of points of complex multiplication is equal to the logarithm of an algebraic number. In this talk I would like to present a proof of this conjecture for a pair of points both lying in the same imaginary quadratic field.

A quantum curve is a magical object. It conjecturally captures information of quantum topological invariants in a concise manner. In this talk we will discuss a method of constructing quantum curves by quantizing the spectral curves of Hitchin systems in terms of the WKB method. The quantization is performed by applying the Eynard-Orantin theory, which is a generalization of the topological recursion formalism developed by Eynard and Orantin in 2007. The talk is based on my joint work with Olivia Dumitrescu.

Let B be a quaternionic algebra over a totally real field F, and p be a prime at least 3 unramified in F. We consider a Shimura variety X associated to B* of level prime to p. A generalization of Deligne-Carayol's "modèle étrange" allows us to define an integral model for X. We will then define a Goren-Oort stratification on the characteristic p fiber of X, and show that each closed Goren-Oort stratum is an iterated P1-fibration over another quaternionic Shimura variety in characteristic p. Now suppose that [F:Q] is even and that p is inert in F. An iteration of this construction gives rise to many algebraic cycles of middle codimension on the characteristic p fibre of Hilbert modular varieties of prime-to-p level. We show that the cohomological classes of these cycles generate a large subspace of the Tate cycles, which, in some special cases, coincides with the prediction of the Tate conjecture for the Hilbert modular variety over finite fields. This is a joint work with Liang Xiao.

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