In these two talks, we consider the Beilinson—Bloch heights and Abel—Jacobian periods of homologically trivial Chow cycles in families. For the Beilinson-Bloch heights, we show that for any $g>2$, there is a non-empty Zariski open subset $U$ of $M_g$, the coarse moduli of curves of genus $g$ over rationals, such that the heights of Ceresa cycles and Gross—Schoen cycles over $U$ satisfy the Northcott property. For the Abel-Jacobi periods, we describe an algebraic criterion for the existence of a non-empty Zariski open subset of any family such that all cycles not defined over $\bar{\mathbb{Q}}$ are non-torsion, and verify that this criterion holds true for Ceresa cycles and Gross—Schoen cycles.

Hain has also proved the second result for the Ceresa cycles using different techniques. He has a conjectural description of the open set for $g=3$.

In the first talk, we explain the arithmetic ingredients, including the Beilinson—Bloch heights, the Ceresa and Gross—Schoen cycles, adelic line bundles, and the arithmetic volume formula. In the second talk, we explain the geometric ingredients, including Griffiths' normal functions, Hain’s metrized biextension line bundles, the criterion of non-degeneracy of Betti maps in this context, and the stratifications of the base variety by algebraic subsets given by Betti foliation.

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