1) We have not yet achieved our overarching goal but we (in work with Laga) have completely resolved our intermediate goal of finding the torsion points on the Jacobians of the curves y^3 = x^4 + ax^2 + b. In subsequent work with Laga, Schembri, and Voight we give nearly optimal constraints on the rational torsion subgroup of *any* abelian surface over Q with (potential) quaternionic multiplication.
2) We have made substantial progress in our approach to the Poonen-Rains heuristics for p = 2 but this is ongoing and we have not released a pre-print yet. Our method allows us to deduce that for any fixed number field F, and for all but at most one quadratic extension K/F, there exists an elliptic curve E/F with positive rank and the same rank over K. Unfortunately, because of the one possible exception, this is not enough to deduce negative solution to Hilbert’s tenth problem over rings of integers. Thus, we found a different way to prove that result (also using Selmer groups, but of odd degree isogenies of CM hyperelliptic Jacbians) in joint work with Alpoge, Bhargava, Ho. There, we prove a variant of the preceding statement with “elliptic curves” replaced by “abelian varieties”, and where no exceptions are allowed.
3) We made much substantial progress in understanding the torsion properties of Ceresa cycles, especially in our two works with Laga (one appearing in Crelle's journal, the other still under review). Namely, we have given many interesting new examples of torsion Ceresa cycles and have shown that their orders in the Chow group are unbounded. A full characterization of torsion Ceresa cycles is still an ambitious goal and will require more work; it is now an even more interesting question, given the exotic examples we've found. Interestingly, these two papers came out of our previous work related to Project 1), which we did not realize would end up being related to the Ceresa cycles of Project 3). In separate work with Lilienfeldt, we have proved a Gross-Zagier type formula for generalized Heegner cycles, which has applications to the motive of Ceresa cycles and their higher dimensional analogues. In the future we hope to drop various assumptions imposed as well as to prove results of this type in a wider context beyond Jacobians of curves with a specific type of complex multiplication.