Explicit methods for rational points on curves and their Jacobians

Projektinformationen

ExplicitRatPoints

ID Finanzhilfevereinbarung: 101064790

DOI

10.3030/101064790

Das Projekt endete am 30 September 2024

EK-Unterschriftsdatum 26 Juli 2022

Startdatum 1 Juni 2023

Enddatum 31 Mai 2025

Finanziert unter

• Marie Skłodowska-Curie Actions (MSCA)

Gesamtkosten

Keine Daten

EU-Beitrag

€ 187 624,32

Koordiniert durch

RIJKSUNIVERSITEIT GRONINGEN
Netherlands

Periodic Reporting for period 1 - ExplicitRatPoints (Explicit methods for rational points on curves and their Jacobians)

Berichtszeitraum: 2023-06-01 bis 2025-05-31

Zusammenfassung vom Kontext und den Gesamtzielen des Projekts

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.

Arbeit, die ab Beginn des Projekts bis zum Ende des durch den Bericht erfassten Berichtszeitraums geleistet wurde, und die wichtigsten bis dahin erzielten Ergebnisse

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.

Fortschritte, die über den aktuellen Stand der Technik hinausgehen und voraussichtliche potenzielle Auswirkungen (einschließlich der bis dato erzielten sozioökonomischen Auswirkungen und weiter gefassten gesellschaftlichen Auswirkungen des Projekts)

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.

I prove theoretical statements that result in algorithms whose output is interesting.