Periodic Reporting for period 3 - UnIntUniBd (Unlikely Intersection and Uniform Bounds for Points)
Reporting period: 2023-04-01 to 2024-09-30
The overall objectives and the problems being addressed of the ERC project are to study and prove uniform bounds in various settings (for example the full Uniform Mordell-Lang Conjecture for all semi-abelian varieties, the Relative Bogomolov Conjecture, Lang's conjecture on uniform bounds of integral points on elliptic curves, etc), some computational aspects, try to get more and more explicit bounds, answer various questions arising in the area of unlikely intersections, and to develop more aspects of unlikely intersections. These results will have applications in other areas than math, for example cryptography.
My group and I addressed many important problems in the area of uniform bounds and on unlikely intersections. They include but are not confined to: (i) Mazur's Conjecture on the rather uniform bound on the number of rational points on curves, which claims that the number of rational points on a smooth projective curve of genus g at least 2 defined over a number field of degree d is bounded above in terms of g, d and the Mordell-Weil rank; (ii) The generic rank of Betti map and the degeneracy loci in families of abelian varieties; (iii) The Uniform Mordell-Lang Conjecture for arbitrary subvarieties of abelian varieties; (iv) The general version of the mixed Ax-Schanuel conjecture for all mixed Shimura varieties, and more generally for all variations of mixed Hodge structures; (v) Equations and multiplicative relations concerning singular moduli; (vi) Development on unlikely intersections in the arithmetic setting; (vii) various questions about torsion points on abelian varieties; (viii) Uniform Mordell-Lang for tori; (ix) Arithmetic bi-Qbar-structure on Shimura varieties; (x) Distinguished categories and Mordell-Lang related questions; (xi) Mahler measure.
Some of these results are the final proofs of some long-open conjectures, for example (i), (iii), (iv) and (vii). Some of these questions are important aspects, tools and cases in studying a major topic (unlikely intersections) of the ERC project, for example (ii), (v), (vi), (viii), (ix), (x) and (xi). These results will have applications in other areas than math, for example cryptography.
Many of these results are presented in various conferences, workshops, seminars, summer schools, etc. I also co-organized a conference.
Some of these results are the final proofs of some long-open conjectures, for example (i), (iii), (iv) and (vii). Some of these questions are important aspects, tools and cases in studying a major topic (unlikely intersections) of the ERC project, for example (ii), (v), (vi), (viii), (ix), (x) and (xi). These results will have applications in other areas than math, for example cryptography.
Many of these results are presented in various conferences, workshops, seminars, summer schools, etc. I also co-organized a conference.
More results from different angles on arithmetic unlikely intersections are expected to be developed. Some developments on Lang's conjecture on the uniform bound of integral points on elliptic curves are expected.