NT seminar tomorrow: Ziyang Gao (UCLA)

[Bianca Viray]

Hi everyone,

Our last NT seminar of the quarter is tomorrow. Our speaker is Ziyang Gao from UCLA and he'll be speaking on Generic positivity of the Beilinson-Bloch height of Gross-Schoen and Ceresa cycles (abstract below). As usual, the talk is at 11am in PDL C-401 and we will go to lunch with the speaker after the talk.

Abstract: Given an algebraic curve defined over a number field, one can define the Néron-Tate height on the Jacobian and prove its positivity. This height pairing and its positivity play important roles in the proof of the Mordell-Weil theorem, in Vojta's proof of the Mordell conjecture, and in the formulation of the BSD conjecture. The Jacobian can be seen, via the Abel-Jacobi map, as the moduli space of 0-cycles of degree 0 on the algebraic curve.

The analogue for higher cycles was studied by Weil, Griffiths, Beilinson, and Bloch. In particular in the 1980s, Beilinson and Bloch independently proposed a conditional definition of heights for arbitrary homologically trivial cycle. The positivity of their heights, as conjectured by Beilinson and Bloch, is widely open.

In this talk, I will report a recent joint work with Shouwu Zhang about a generic positivity for the Gross-Schoen and Ceresa cycles of curves of genus at least 3. These are the simplest situation where the Beilinson-Bloch heights are unconditionally defined.

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Bianca Viray (she/her)

Craig McKibben and Sarah Merner Professor of Mathematics

University of Washington

Seattle, WA 98195

https://sites.math.washington.edu/~bviray/