I made progress on the Birch–Swinnerton-Dyer conjecture in three different settings:
1. I verified the strong form of the conjecture for the first time for many non-trivial cases of "abelian surfaces" (analogs of elliptic curves of dimension 2) over the rationals. This was possible through theoretical and algorithmic advances.
2. a) Using Iwasawa theory, I proved the p-part of the strong BSD conjecture over Q in analytic rank 0 and 1 for odd Eisenstein primes. This is the first general result in the case the elliptic curve has non-trivial p-torsion. b) We proved a p-converse theorem for odd additive Eisenstein primes of good ordinary reduction, allowing us, for example, to deduce better proportions for Goldfeld's conjecture for elliptic curves having reducible mod-3 Galois representation.
3. I proved the equivalence of the BSD conjecture with the ell- or p-part of the Tate–Shafarevich group having finite exponent over higher-dimensional function fields in positive characteristic. The equivalent conditions are algorithmically verifyable in principle.
I made progress on rational points on modular curves in the following cases:
1. An algorithm and a precise conjecture on sporadic points on modular curves.
2. Elkies' conjecture for the modular curve X_0(N)^*: For N non-squarefree we made important steps towards a full proof. For N squarefee we are determining the values of the local heights needed by the Quadratic Chabauty method.
3. In a project on the Inverse Galois Problem for the group 17T7, we worked heavily with Shimura curves, a generalization of modular curves.
