Monodromy of the Kodaira--Parshin family

Abstract: We complete the final step of the proof, proving that the monodromy of the Kodaira-Parshin family is large. Let $Y$ be a compact orientable surface or genus $g\ge 2$ and let $Y^0$ be the puncture at a point. Let $(Z_1,\pi_1),\dots,(Z_N,\pi_N)$ be the $\mathrm{Aff}(q)$-covers of $Y^0$, and let $\mathrm{MCG}(Y^0)_0$ be those mapping classes of $Y^0$ who induce trivial mapping classes on every $Z_i$. Then $\mathrm{MCG}(Y^0)_0$ acts on the primitive homomology $H_1^{\mathrm{Pr}}(Z_i,Y^0)$ of $Z_i$, i.e., the orthogonal complement of $\pi_i^*H_1(Z_i)\subset H_1(Y^0)$. We prove that $\mathrm{MCG}(Y^0)_0\to\prod_{i=1}^N\mathrm{Sp}(H_1^{\mathrm{Pr}}(Z_i,Y^0))$ has Zariski dense image.

By Goursat's lemma it suffices to prove each $\mathrm{MCG}(Y^0)_0\to\mathrm{Sp}(H_1^{\mathrm{Pr}}(Z_i,Y^0))$ has Zariski dense image, which we prove by producing many Dehn twists on $Y^0$ inducing trivial mapping classes on $Z_i$.

Reference:

$\bullet$ Lawrence and Venkatesh, Diophantine problems and $p$-adic period mappings, Section 8.

algebraic geometrynumber theory

Audience: advanced learners

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STAGE

Series comments: STAGE (Seminar on Topics in Arithmetic, Geometry, Etc.) is a learning seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.

Fall 2025 topic: Weil conjectures.

Some topics might take more or less time than allotted. If a speaker runs out of time on a certain date, that speaker might be allowed to borrow some time on the next date. So the topics below might not line up exactly with the dates below.

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