‎‎In the category of Mordell curves (E_D:y^2=x^3+D) with nontrivial torsion groups we find curves of the generic rank two as quadratic twists of (E_1), ‎and of the generic rank at least two and at least three as cubic twists of (E_1). ‎Previous work‎, ‎in the category of Mordell curves with trivial torsion groups‎, ‎has found infinitely many elliptic curves with ...

We consider the local-global principle for divisibility in Mordell-Weil group of a CM elliptic curve defined over number field. For each prime p we give lower bounds on degree d field which there exists gives counterexample to by power p. As corollary deduce that are at most finitely many curves (with or without CM) counterexamples with p>2d+1. also powers 7 holds all quadratic fields.

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for prime number p, in the p-adic Lie group jacobian, closure Mordell-Weil with curve. If rank less than then this has never failed. Minhyong Kim's non-abelian Chabauty programme aims remove condition rank. The simplest case, called quadratic Chabauty, was developed by Bala...

We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the ...

Let C be a smooth projective absolutely irreducible curve of genus g ≥ 2 over a number field K, and denote its Jacobian by J . Let d ≥ 1 be an integer and denote the d-th symmetric power of C by C(d). In this paper we adapt the classic Chabauty–Coleman method to study the K-rational points of C(d). Suppose that J(K) has Mordell–Weil rank at most g− d. We give an explicit and practical criterion...

We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus > 1, it is impractical to apply Hilbert’s Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrar...

Periodic points are points on Veech surfaces, whose orbit under the group of affine diffeomorphisms is finite. We characterise those points as being torsion points if the Veech surfaces is suitably mapped to its Jacobian or an appropriate factor thereof. For a primitive Veech surface in genus two we show that the only periodic points are the Weierstraß points and the singularities. Our main too...

‎‎in the category of mordell curves (e_d:y^2=x^3+d) with nontrivial torsion groups we find curves of the generic rank two as quadratic twists of (e_1), ‎and of the generic rank at least two and at least three as cubic twists of (e_1). ‎previous work‎, ‎in the category of mordell curves with trivial torsion groups‎, ‎has found infinitely many elliptic curves with rank at least seven as sextic tw...

Let [Formula: see text] be an elliptic curve defined over a number field where splits completely. Suppose that has good reduction at all primes above text]. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups the cyclotomic text]-extension finite extension is unramified. Under hypothesis Pontryagin duals these are torsion corresponding Iwasawa algebra, s...

We explicitly define the set of algebraic points a hyperelliptic curve $\mathcal{C}_q$ any given degree having affine equation $y^2=(x-2 q)\left(x^2-2 q^2\right)\left(x^2+2^2 q^2\right)$. Such is described in [4], where it shown that Mordell-Weil group finite for $q \equiv 13$ [24]. Furthermore, generators torsion these curves are explained. Received: March 25, 2023Accepted: July 3, 2023