Mordell-Weil Rank Research Articles

Non-thin rank jumps for double elliptic K3 surfaces

AbstractFor an elliptic surface $$\pi :X\rightarrow \mathbb {P}^1$$ π : X → P 1 defined over a number field K, a theorem of Silverman shows that for all but finitely many fibres above K-rational points, the resulting elliptic curve over K has Mordell-Weil rank at least as large as the rank of the group of sections of $$\pi $$ π . When X is a K3 surface with two distinct elliptic fibrations, we show that the set of K-rational points of $$\mathbb {P}^1$$ P 1 for which this rank inequality is strict, is not a thin set, under certain hypothesis on the fibrations. Our results provide one of the first cases of this phenomenon beyond that of rational elliptic surfaces.

A function related to the Mordell–Weil rank of elliptic curves

Let p be a prime number. For each natural number n, we study the behavior of the function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square (mod p). The study of this function is inspired by the cognate function [Formula: see text] which enumerates the number of factorizations [Formula: see text] with [Formula: see text] a perfect square. The descent theory of elliptic curves would show that if [Formula: see text] is unbounded for squarefree values of n, then there are elliptic curves over the rational number field with arbitrarily large rank. In this note, we show for every prime p, [Formula: see text] is unbounded as n ranges over squarefree values, thus providing some evidence for the conjecture that [Formula: see text] is unbounded for squarefree n.

Expected Mordell-Weil rank heuristics through Sato-Tate, Birch and Swinnerton-Dyer conjectures

Expected Mordell-Weil rank heuristics through Sato-Tate, Birch and Swinnerton-Dyer conjectures

Uniformity of quadratic points

In this paper, we extend a uniformity result of Dimitrov et al. [Uniformity in Mordell-Lang for curves, Ann. of Math. (2) 194(1) (2021) 237–298] to dimension two and use it to get a uniform bound on the cardinality of the set of all quadratic points for non-hyperelliptic non-bielliptic curves which only depend on the Mordell–Weil rank, the genus of the curve and the degree of the number field.

Specializations of a generic rank-2 curve of Shioda

In a paper from twenty years ago, Brown and Myers proved the clean and uniform result that if m>1 is an integer, then every curve of the form y2=x3−x+m2 has Mordell-Weil rank at least 2. In this article, we further our investigation into the more general family of curves, in which the polynomial x3−x is replaced by an arbitrary cubic which is squarefree and has three integral roots. In particular, a new strategy will unveil a subfamily of curves with surprisingly large average rank, and one startling example.

Dynamics of quadratic polynomials and rational points on a curve of genus 4

Let f t ( z ) = z 2 + t f_t(z)=z^2+t . For any z ∈ Q z\in \mathbb {Q} , let S z S_z be the collection of t ∈ Q t\in \mathbb {Q} such that z z is preperiodic for f t f_t . In this article, assuming a well-known conjecture of Flynn, Poonen, and Schaefer [Duke Math. J. 90 (1997), pp. 435–463], we prove a uniform result regarding the size of S z S_z over z ∈ Q z\in \mathbb {Q} . In order to prove it, we need to determine the set of rational points on a specific non-hyperelliptic curve C C of genus 4 4 defined over Q \mathbb {Q} . We use Chabauty’s method, which requires us to determine the Mordell-Weil rank of the Jacobian J J of C C . We give two proofs that the rank is 1 1 : an analytic proof, which is conditional on the BSD rank conjecture for J J and some standard conjectures on L-series, and an algebraic proof, which is unconditional, but relies on the computation of the class groups of two number fields of degree 12 12 and degree 24 24 , respectively. We finally combine the information obtained from both proofs to provide a numerical verification of the strong BSD conjecture for J J .

Geometric quadratic Chabauty and [formula omitted]-adic heights

Let X be a curve of genus g>1 over Q whose Jacobian J has Mordell–Weil rank r and Néron–Severi rank ρ. When r<g+ρ−1, the geometric quadratic Chabauty method determines a finite set of p-adic points containing the rational points of X. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of p-adic heights and p-adic (Coleman) integrals. This translation also allows us to give a comparison to the (original) cohomological method for quadratic Chabauty. We show that the finite set of p-adic points produced by the geometric method is contained in the finite set produced by the cohomological method, and give a description of their difference.

Quadratic Chabauty for modular curves: algorithms and examples

We describe how the quadratic Chabauty method may be applied to determine the set of rational points on modular curves of genus $g&gt;1$ whose Jacobians have Mordell–Weil rank $g$. This extends our previous work on the split Cartan curve of level 13 and allows us to consider modular curves that may have few known rational points or non-trivial local height contributions at primes of bad reduction. We illustrate our algorithms with a number of examples where we determine the set of rational points on several modular curves of genus 2 and 3: this includes Atkin–Lehner quotients $X_0^+(N)$ of prime level $N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's Program B. We also compute the set of rational points on the genus 6 non-split Cartan modular curve $X_{\scriptstyle \mathrm { ns}} ^+ (17)$.

Rank growth of elliptic curves over 𝑁-th root extensions

Fix an elliptic curve E E over a number field F F and an integer n n which is a power of 3 3 . We study the growth of the Mordell–Weil rank of E E after base change to the fields K d = F ( d 2 n ) K_d = F(\!\sqrt [2n]{d}) . If E E admits a 3 3 -isogeny, then we show that the average “new rank” of E E over K d K_d , appropriately defined, is bounded as the height of d d goes to infinity. When n = 3 n = 3 , we moreover show that for many elliptic curves E / Q E/\mathbb {Q} , there are no new points on E E over Q ( d 6 ) \mathbb {Q}(\sqrt [6]d) , for a positive proportion of integers d d . This is a horizontal analogue of a well-known result of Cornut and Vatsal [Nontriviality of Rankin-Selberg L-functions and CM points, L-functions and Galois representations, vol. 320, Cambridge Univ. Press, Cambridge, 2007, pp. 121–186]. As a corollary, we show that Hilbert’s tenth problem has a negative solution over a positive proportion of pure sextic fields Q ( d 6 ) \mathbb {Q}(\sqrt [6]{d}) . The proofs combine our recent work on ranks of abelian varieties in cyclotomic twist families with a technique we call the “correlation trick”, which applies in a more general context where one is trying to show simultaneous vanishing of multiple Selmer groups. We also apply this technique to families of twists of Prym surfaces, which leads to bounds on the number of rational points in sextic twist families of bielliptic genus 3 curves.

Specialization of Mordell–Weil ranks of abelian schemes over surfaces to curves

Using the Shioda–Tate theorem and an adaptation of Silverman’s specialization theorem, we reduce the specialization of Mordell–Weil ranks for abelian varieties over fields finitely generated over infinite finitely generated fields [Formula: see text] to the specialization theorem for Néron–Severi ranks recently proved by Ambrosi in positive characteristic. More precisely, we prove that after a blow-up of the base surface [Formula: see text], for all vertical curves [Formula: see text] of a fibration [Formula: see text] with [Formula: see text] from the complement of a sparse subset of [Formula: see text], the Mordell–Weil rank of an abelian scheme over [Formula: see text] stays the same when restricted to [Formula: see text].

Determination of the modular Jacobian varieties J_1(M,MN) with the Mordell–Weil rank zero

In this paper, we determine all modular Jacobian varieties J_1(M,MN) over the cyclotomic fields {{mathbb {Q}}}(zeta _M) with the Mordell–Weil rank zero assuming the Birch–Swinnerton-Dyer conjecture, following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.

Twists of Albanese varieties of abelian and dihedral covers of ℙℓ with large ranks over function fields

In this paper, we construct twists of Albanese varieties of abelian and dihedral covers of projective spaces with large Mordell–Weil rank over function fields of certainly defined high dimensional varieties. We achieve this by using the theory of twists, a generalization of the notion of Prym variety to higher dimensions, and a structure theorem for the Mordell–Weil group of abelian varieties over function fields.

A note on rank one quadratic twists of elliptic curves and the non-degeneracy of 𝑝-adic regulators at Eisenstein primes

We show that for certain non-CM elliptic curves E / Q E_{/\mathbb {Q}} such that 3 3 is an Eisenstein prime of good reduction, a positive proportion of the quadratic twists E ψ E_{\psi } of E E have Mordell–Weil rank one and the 3 3 -adic height pairing on E ψ ( Q ) E_{\psi }(\mathbb {Q}) is non-degenerate. We also show similar but weaker results for other Eisenstein primes. The method of proof also yields examples of middle codimensional algebraic cycles over number fields of arbitrarily large dimension (generalized Heegner cycles) that have non-zero p p -adic height. It is not known – though expected – that the archimedian height of these higher-codimensional cycles is non-zero.

Geometric quadratic Chabauty over number fields

This article generalizes the geometric quadratic Chabauty method, initiated over Q \mathbb {Q} by Edixhoven and Lido, to curves defined over arbitrary number fields. The main result is a conditional bound on the number of rational points on curves that satisfy an additional Chabauty type condition on the Mordell–Weil rank of the Jacobian. The method gives a more direct approach to the generalization by Dogra of the quadratic Chabauty method to arbitrary number fields.

ON THE FIBRES OF AN ELLIPTIC SURFACE WHERE THE RANK DOES NOT JUMP

AbstractFor a nonconstant elliptic surface over $\mathbb {P}^1$ defined over $\mathbb {Q}$ , it is a result of Silverman [‘Heights and the specialization map for families of abelian varieties’, J. reine angew. Math.342 (1983), 197–211] that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is nonisotrivial, one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann [‘Elliptische Kurven mit vorgeschriebenem Reduktionsverhalten. I’, Math. Nachr.49 (1971), 107–123] and Setzer [‘Elliptic curves of prime conductor’, J. Lond. Math. Soc. (2)10 (1975), 367–378]. In this note, we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.

On the rank of cubic and quartic twists of elliptic curves by primes

We show that there are infinitely many primes p for which the cubic twist y2=x3+16p2 (respectively the quartic twist y2=x3−px) has positive Mordell-Weil rank by playing off the structure of the Q-rational torsion points again the projective geometry of these twists. A key ingredient to both proofs are sieve results of Heath-Brown and Moroz, and Friedlander and Iwaniec, on primes represented by cubic and quartic polynomials. As a consequence we show that there are infinitely many primes congruent to 1(mod9) and respectively −1(mod9) that can be written as a sum of two rational cubes. We show that there are infinitely many primes p for which the cubic twist y2=x3+16p2 (respectively the quartic twist y2=x3−px) has positive Mordell-Weil rank by playing off the structure of the Q-rational torsion points again the projective geometry of these twists. A key ingredient to both proofs are sieve results of Heath-Brown and Moroz, and Friedlander and Iwaniec, on primes represented by cubic and quartic polynomials. As a consequence we show that there are infinitely many primes congruent to 1(mod9) and respectively −1(mod9) that can be written as a sum of two rational cubes.

Explicit two-cover descent for genus 2 curves

Given a genus 2 curve C with a rational Weierstrass point defined over a number field, we construct a family of genus 5 curves that realize descent by maximal unramified abelian two-covers of C, and describe explicit models of the isogeny classes of their Jacobians as restrictions of scalars of elliptic curves. All the constructions of this paper are accompanied by explicit formulas and implemented in Magma and/or SageMath. We apply these algorithms in combination with elliptic Chabauty to a dataset of 7692 genus 2 quintic curves over $$\mathbb {Q}$$ of Mordell–Weil rank 2 or 3 whose sets of rational points have not previously been provably computed. We analyze how often this method succeeds in computing the set of rational points and what obstacles lead it to fail in some cases.

REMARKS ON HILBERT’S TENTH PROBLEM AND THE IWASAWA THEORY OF ELLIPTIC CURVES

AbstractLet E be an elliptic curve with positive rank over a number field K and let p be an odd prime number. Let $K_{\operatorname {cyc}}$ be the cyclotomic $\mathbb {Z}_p$ -extension of K and $K_n$ its nth layer. The Mordell–Weil rank of E is said to be constant in the cyclotomic tower of K if for all n, the rank of $E(K_n)$ is equal to the rank of $E(K)$ . We apply techniques in Iwasawa theory to obtain explicit conditions for the rank of an elliptic curve to be constant in this sense. We then indicate the potential applications to Hilbert’s tenth problem for number rings.

Rational points on quadratic elliptic surfaces

We consider elliptic surfaces whose coefficients are degree 2 polynomials in a variable T. It was recently shown, Kollár and Mella (Amer J Math 139(4):915–936 2017), that for infinitely many rational values of T the resulting elliptic curves have rank at least 1. We prove that the Mordell–Weil rank of each such elliptic surface is at most 6 over \({{\mathbb {Q}}}\). In fact, we show that the Mordell–Weil rank of these elliptic surfaces is controlled by the number of zeros of a certain polynomial over \({{\mathbb {Q}}}\).

Rank jumps on elliptic surfaces and the Hilbert property

Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.