## Fall 2022 @ Amherst College - September 30, 2022

The BATMOBILE is a vehicle for bringing together the algebraic and
tropical geometry communities of Brown and surrounding institutions
for a biannual day of talks.

Abstracts

Padma Srinivasan (ICERM) -- Towards a unified theory of canonical heights on abelian varieties

*p*-adic heights have been a rich source of explicit functions vanishing on rational points on a curve. In this talk, we will outline a new construction of canonical *p*-adic heights on abelian varieties from *p*-adic adelic metrics, using *p*-adic Arakelov theory developed by Besser. This construction closely mirrors Zhang's construction of canonical real valued heights from real-valued adelic metrics. We will use this new construction to give direct explanations (avoiding *p*-adic Hodge theory) of the key properties of height pairings needed for the quadratic Chabauty method for rational points. This is joint work with Amnon Besser and Steffen Mueller.

Sam Payne (UT Austin) -- Cohomology of moduli spaces of curves

The Langlands program makes striking predictions about the Hodge structures and Galois representations that can appear in the cohomology of moduli spaces of curves, via the conjectured correspondence with automorphic cuspidal representations of conductor 1 that are now classified in weights up to 22, by Chenevier and Lannes. In this talk, I will outline the main ideas underlying these predictions and present joint work from two recent projects, one joint with Jonas BergstrÃ¶m and Carel Faber, and the other joint with Sam Canning and Hannah Larson, that confirm these predictions in weights up to 12, and also provide a surprisingly precise description of the eleventh cohomology group of the Deligne-Mumford moduli space of stable curves.

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