Tuesday, July 19, 2022 at 12:00 p.m.

PhD Candidate: Georgios Papas

Supervisor: Jacob Tsimerman

Thesis title: Some topics in the arithmetic of Hodge structures and an Ax-Scanuel theorem for GLn

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In the first part of this thesis we consider smooth projective morphisms $f:X\rightarrow S$ of $K$-varieties with $S$ an open curve and $K$ a number field. We establish upper bounds of the Weil height $h(s)$ by $[K(s):K]$ at certain points $s\in S(\bar{K})$ that are “exceptional” with respect to the variation of Hodge structures $R^n(f^{an})_{*}(\Q_{X^{an}_{\C}})$, where $n=\dim X-1$. We work under the assumption that the generic special Mumford-Tate group of this variation is $Sp(\mu,\Q)$, the variation degenerates in a strong fashion over some fixed point $s_0$ of a proper curve that contains $S$, the Hodge conjecture holds, and that what we define as a “good arithmetic model” exists for the morphism $f$ over the ring $\mathcal{O}_K$.

Our motivation comes from the field of unlikely intersections, where analogous bounds were used to settle unconditionally certain cases of the Zilber-Pink conjecture.

In the second part of this thesis, we prove an Ax-Schanuel type result for the exponential functions of general linear groups over $\mathbb{C}$. We prove the result first for the group of upper triangular matrices and then for the group $GL_n$ of all $n\times n$ invertible matrices over $\mathbb{C}$. We also obtain Ax-Lindemann type results for these maps as a corollary, characterizing the bi-algebraic subsets of these maps.

Our motivation comes from the fact that Ax-Schanuel and Ax-Lindemann type results are an important tool in the theory of unlikely intersections, in the context of the Pila-Zannier method.

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A draft of the thesis can be found here: G.Papas, Thesis