Periodic Reporting for period 1 - ExplicitRatPoints (Explicit methods for rational points on curves and their Jacobians)
Reporting period: 2023-06-01 to 2025-05-31
The study of diophantine equations, polynomial equations like x³ + y³ = z³ where only integer or rational solutions are sought for, has a very long history. Cubic equations like y² = x³ + 1 are particularly interesting because their solutions have additional structure. Still, there is no algorithm that always "computes" (in a certain sense because there can be infinitely many solutions) their solutions. The conjecture of Birch–Swinnerton-Dyer gives a conjectural solution to this problem. My research is concerned with a stronger version of that conjecture in three different settings (positive characteristic, algorithmic and theoretical over the rationals).
These cubic equations are "classfied" by other equations, whose associated solution set is a modular curve. Determining their solutions is an active area of research. I work and worked on several projects towards a deeper understanding on them.
These cubic equations are "classfied" by other equations, whose associated solution set is a modular curve. Determining their solutions is an active area of research. I work and worked on several projects towards a deeper understanding on them.
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.
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.
Regarding BSD:
1. I'm working on the verification over totally real fields and in dimension 3.
2. I'm working on p-converse theorems for odd Eisenstein primes of bad, potentially multiplicative and potentially supersingular reduction.
3. I'm working on a strong BSD formula over higher-dimensional function fields in positive characteristic.
Regarding modular curves:
1. We are working on completing our proof for Elkies' conjecture on X_0(N)^*(Q) for N non-squarefree. We expect an article in 2025.
2. We are working on a modular algorithm to compute the local heights away from p for X_0(N)^* for N squarefree. We expect an article in 2025.
1. I'm working on the verification over totally real fields and in dimension 3.
2. I'm working on p-converse theorems for odd Eisenstein primes of bad, potentially multiplicative and potentially supersingular reduction.
3. I'm working on a strong BSD formula over higher-dimensional function fields in positive characteristic.
Regarding modular curves:
1. We are working on completing our proof for Elkies' conjecture on X_0(N)^*(Q) for N non-squarefree. We expect an article in 2025.
2. We are working on a modular algorithm to compute the local heights away from p for X_0(N)^* for N squarefree. We expect an article in 2025.