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Number Theory Seminar

Title: Moduli space of abelian surfaces with fixed $3$-torsion representation
Speaker: Shiva Chidambaram (MIT, USA)
Date: 07 September 2022
Time: 11.30 AM
Venue: LH-3

The modularity lifting theorem of Boxer-Calegari-Gee-Pilloni established for the first time the existence of infinitely many modular abelian surfaces $A / \mathbb{Q}$ upto twist with $\text{End}_{\mathbb{C}}(A) = \mathbb{Z}$. We render this explicit by first finding some abelian surfaces whose associated mod-$p$ representation is residually modular and for which the modularity lifting theorem is applicable, and then transferring modularity in a family of abelian surfaces with fixed $3$-torsion representation. Let $\rho: G_{\mathbb{Q}} \rightarrow GSp(4,\mathbb{F}_3)$ be a Galois representation with cyclotomic similitude character. Then, the transfer of modularity happens in the moduli space of genus $2$ curves $C$ such that $C$ has a rational Weierstrass point and $\mathrm{Jac}(C)[3] \simeq \rho$. Using invariant theory, we find explicit parametrization of the universal curve over this space. The talk will feature demos of relevant code in Magma.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 08 Dec 2022