Fix a non-square integer $$k\ne 0$$ . We show that the number of curves $$E_B:y^2=x^3+kB^2$$ containing an integral point, where B ranges over positive integers less than N, is bounded by $$\ll _k N(\log N)^{-\frac{1}{2}+\epsilon }$$ In particular, this implies $$B\le N$$ such $$-3kB^2$$ discriminant elliptic curve $$\mathbb {Q}$$ o(N). The proof involves discriminant-lowering procedure on bina...

Chromatic Selmer groups are modified with local information for supersingular primes p. We sketch their role in establishing the p-primary part of Birch–Swinnerton-Dyer formula Sections 2–5, and then study growth Mordell–Weil rank along ℤ p 2 -extension a quadratic imaginary number field which splits Section 6.

For modular elliptic curves over number fields of narrow class one, and with multiplicative reduction at a collection p-adic primes, we define new invariants. Inspired by Nekovář Scholl's plectic conjectures, believe these invariants control the Mordell–Weil group higher rank support our expectations numerical experiments.

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to question Mazur's concerning uniformity Mordell–Lang for curves. In particular we implies uniform Manin–Mumford The proof is built up on our previous work [DGH21].

We refine necessary and sufficient conditions for the generating series of a weighted model quarter plane walk to be differentially algebraic. In addition, we give algorithms based on theory Mordell–Weil lattices, that, each model, yield polynomial weights determining this property associated series.

In this paper we generalize the known De Smet, Dillen, Verstraelen and Vrancken (DDVV)-type inequalities for real (skew-)symmetric complex (skew-)Hermitian matrices to arbitrary real, quaternionic matrices. Inspired by Erdős-Mordell inequality, establish DDVV-type in subspaces spanned a Clifford system or algebra. We also Böttcher-Wenzel inequality

The theory of Mordell-Weil lattices is applied to a specific example of a rational elliptic surface. This provides a complete description of the sections of this surface, and of the sections which are defined over Q. The surface is related to the diophantine problem of expressing squares as a sum of consecutive squares. Some consequences which our description has to this problem are discussed.

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman’s the...

A polynomial relation f(x, y) = 0 in two variables defines a curve C. If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f(x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. If we consider a non-singular projective model C of the curve then over C it is classified by its genus. Mordell conjectured, and in 1983 Falting...