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.

- All seminars.
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Last updated: 08 Oct 2024