Mordell-Weil Rank Research Articles

On the Rank of Elliptic Curves in Elementary Cubic Extensions

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

On the class numbers of the fields of the [formula omitted]-torsion points of certain elliptic curves over [formula omitted

Let E be an elliptic curve over Q with prime conductor p. For each positive integer n we put Kn:=Q(E[pn]). The aim of this paper is to estimate the order of the p-Sylow group of the ideal class group of Kn. We give a lower bound in terms of the Mordell–Weil rank of E(Q). As an application of our result, we give an example such that p2n divides the class number of the field Kn in the case of p=5077 for each positive integer n.

On the ranks of elliptic curves in families of quadratic twists over number fields

A conjecture due to Honda predicts that given any abelian variety over a number field K, all of its quadratic twists (or twists of a fixed order in general) have bounded Mordell-Weil rank. About 15 years ago, Rubin and Silverberg obtained an analytic criterion for Honda’s conjecture for a family of quadratic twists of an elliptic curve defined over the field of rational numbers. In this paper, we consider this problem over number fields. We will prove that the existence of a uniform upper bound for the ranks of elliptic curves in this family is equivalent to the convergence of a certain infinite series. This result extends the work of Rubin and Silverberg over ℚ.

Quadratic Chabauty:p-adic heights and integral points on hyperelliptic curves

AbstractWe give a formula for the component atpof thep-adic height pairing of a divisor of degree 0 on a hyperelliptic curve. We use this to give a Chabauty-like method for findingp-adic approximations top-integral points on such curves when the Mordell–Weil rank of the Jacobian equals the genus. In this case we get an explicit bound for the number of suchp-integral points, and we are able to use the method in explicit computation. An important aspect of the method is that it only requires a basis of the Mordell–Weil group tensored withℚ$\mathbb{Q}$.

F-GUTs with Mordell–Weil [formula omitted]'s

In this note we study the constraints on F-theory GUTs with extra U(1)'s in the context of elliptic fibrations with rational sections. We consider the simplest case of one abelian factor (Mordell–Weil rank one) and investigate the conditions that are induced on the coefficients of its Tate form. Converting the equation representing the generic hypersurface P112 to this Tate's form we find that the presence of a U(1), already in this local description, is consistent with the exceptional E6 and E7 non-abelian singularities. We briefly comment on a viable E6×U(1) effective F-theory model.

The p-parity conjecture for elliptic curves with a p-isogeny

Abstract For an elliptic curve E over a number field K, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell–Weil rank. Assuming finiteness of Ш ⁢ ( E / K ) ⁢ [ p ∞ ] ${\Sha(E/K)[p^{\infty}]}$ for a prime p this is equivalent to the p-parity conjecture: the global root number matches the parity of the ℤ p ${\mathbb{Z}_{p}}$ -corank of the p ∞ ${p^{\infty}}$ -Selmer group. We complete the proof of the p-parity conjecture for elliptic curves that have a p-isogeny for p > 3 ${p>3}$ (the cases p ≤ 3 ${p\leq 3}$ were known). Tim and Vladimir Dokchitser have showed this in the case when E has semistable reduction at all places above p by establishing respective cases of a conjectural formula for the local root number. We remove the restrictions on reduction types by proving their formula in the remaining cases. We apply our result to show that the p-parity conjecture holds for every E with complex multiplication defined over K. Consequently, if for such an elliptic curve Ш ⁢ ( E / K ) ⁢ [ p ∞ ] ${\Sha(E/K)[p^{\infty}]}$ is infinite, it must contain ( ℚ p / ℤ p ) 2 ${(\mathbb{Q}_{p}/\mathbb{Z}_{p})^{2}}$ .

Hodge theory and the Mordell–Weil rank of elliptic curves over extensions of function fields

We use Hodge theory to prove a new upper bound on the ranks of Mordell–Weil groups for elliptic curves over function fields after regular geometrically Galois extensions of the base field, improving on previous results of Silverman and Ellenberg, when the base field has characteristic zero and the supports of the conductor of the elliptic curve and of the ramification divisor of the extension are disjoint.

Explicit Chabauty over number fields

Let C be a smooth projective absolutely irreducible curve of genus g≥2 over a number field K of degree d, and let J denote its Jacobian. Let r denote the Mordell–Weil rank of J(K). We give an explicit and practical Chabauty-style criterion for showing that a given subset K⊆C(K) is in fact equal to C(K). This criterion is likely to be successful if r≤d(g−1). We also show that the only solution to the equation x2+y3=z10 in coprime nonzero integers is (x,y,z)=(±3,−2,±1). This is achieved by reducing the problem to the determination of K-rational points on several genus-2 curves where K=ℚ or ℚ(23) and applying the method of this paper.

On Mordell–Weil groups of isotrivial abelian varieties over function fields

We show that the Mordell–Weil rank of an isotrivial abelian variety with cyclic holonomy depends only on the fundamental group of the complement to the discriminant, provided the discriminant has singularities in CM class introduced here. This class of singularities includes all unibranched plane curves singularities. As a corollary, we describe a family of simple Jacobians over the field of rational functions in two variables for which the Mordell–Weil rank is arbitrarily large.

Mordell–Weil ranks of families of elliptic curves associated to Pythagorean triples

We study the family of elliptic curves $y^2=x(x-a^2)(x-b^2)$ parametrized by Pythagorean triples $(a,b,c)$. We prove that for a generic triple the lower bound of the rank of the Mordell-Weil group over $\mathbb{Q}$ is 1, and for some explicitly given infinite family the rank is 2. To each family we attach an elliptic surface fibered over the projective line. We show that the lower bounds for the rank are optimal, in the sense that for each generic fiber of such an elliptic surface its corresponding Mordell-Weil group over the function field $\mathbb{Q}(t)$ has rank 1 or 2, respectively. In order to prove this, we compute the characteristic polynomials of the Frobenius automorphisms acting on the second $\ell$-adic cohomology groups attached to elliptic surfaces of Kodaira dimensions 0 and 1.

The Rank of Jacobian Varieties over the Maximal Abelian Extensions of Number Fields: Towards the Frey–Jarden Conjecture

AbstractFrey and Jarden asked if any abelian variety over a number field K has the infinite Mordell–Weil rank over the maximal abelian extension Kab. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve C over K such that #C(Kab) = ∞ and for any abelian variety of GL2-type with trivial character.

Cuspidal plane curves, syzygies and a bound on the MW-rank

Let C=Z(f) be a reduced plane curve of degree 6k, with only nodes and ordinary cusps as singularities. Let I be the ideal of the points where C has a cusp. Let ⊕S(−bi)→⊕S(−ai)→S→S/I be a minimal resolution of I. We show that bi⩽5k. From this we obtain that the Mordell–Weil rank of the elliptic threefold W:y2=x3+f equals 2#{i|bi=5k}. Using this we find an upper bound for the Mordell–Weil rank of W, which is 118(125+73−2302−10673)k+l.o.t. and we find an upper bound for the exponent of (t2−t+1) in the Alexander polynomial of C, which is 136(125+73−2302−10673)k+l.o.t. This improves a recent bound of Cogolludo and Libgober almost by a factor 2.

The maximal rank of elliptic Delsarte surfaces

Shioda described a method to compute the Lefschetz number of a Delsarte surface. In one of his examples he uses this method to compute the rank of an elliptic curve over k(t). In this article we find all elliptic curves over k(t) for which his method is applicable. For these curves we also compute the maximal Mordell-Weil rank.

A note on the Mordell–Weil rank modulo n

Conjecturally, the parity of the Mordell–Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is (conjecturally) the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n > 2 , so this is purely a parity phenomenon.

Nonvanishing of Hecke L-Functions for CM Fields and Ranks of Abelian Varieties

In this paper we prove a nonvanishing theorem for central values of L-functions associated to a large class of algebraic Hecke characters of CM number fields. A key ingredient in the proof is an asymptotic formula for the average of these central values. We combine the nonvanishing theorem with work of Tian and Zhang [TiZ] to deduce that infinitely many of the CM abelian varieties associated to these Hecke characters have Mordell–Weil rank zero. Included among these abelian varieties are higher-dimensional analogues of the elliptic $${{\mathbb Q}}$$ -curves A(D) of B. Gross [Gr].

On quadratic families of CM elliptic curves

Given a CM elliptic curve with Weierstrass equation y 2 = f(x), and a positive definite binary quadratic form Q(v, v), we show that there are infinitely many reduced integer pairs (u, v) such that the twisted elliptic curve Q(u, v)y 2 = f(x) has analytic rank (and consequently Mordell-Weil rank) one. In fact it follows that the number of such pairs with |u|, |v| ≤ X is at least X 2―e for any e > 0.

Calculating the Mordell-Weil rank of elliptic threefolds and the cohomology of singular hypersurfaces

In this paper we give a method for calculating the rank of a general elliptic curve over the field of rational functions in two variables. We reduce this problem to calculating the cohomology of a singular hypersurface in a weighted projective 4-space. We then give a method for calculating the cohomology of a certain class of singular hypersurfaces, extending work of Dimca for the isolated singularity case.

Extremal rational elliptic threefolds

An elliptic fibration is a proper morphism f : X → Y of normal projective varieties whose generic fibre E is a regular curve of genus 1. The Mordell–Weil rank of such a fibration is defined to be the rank of the finitely generated abelian group Pic0 E of degree-0 line bundles on E. In particular, f is called extremal if its Mordell–Weil rank is 0. The simplest nontrivial elliptic fibration is a rational elliptic surface f : X → P1. There is a complete classification of extremal rational elliptic surfaces, due to Miranda–Persson in characteristic 0 [14] and W. Lang in positive characteristic [12, 13]. (See also Cossec– Dolgachev [4, Section 5.6].) The purpose of the present paper is to produce a corresponding classification of a certain class of extremal rational elliptic threefolds.

On the Birch-Swinnerton-Dyer quotients modulo squares

Let A be an abelian variety over a number field K. An identity between the L-functions L(A/K_i,s) for extensions K_i of K induces a conjectural relation between the Birch-Swinnerton-Dyer quotients. We prove these relations modulo finiteness of Sha, and give an analogous statement for Selmer groups. Based on this, we develop a method for determining the parity of various combinations of ranks of A over extensions of K. As one of the applications, we establish the parity conjecture for elliptic curves assuming finiteness of Sha[6^\infty] and some restrictions on the reduction at primes above 2 and 3: the parity of the Mordell-Weil rank of E/K agrees with the parity of the analytic rank, as determined by the root number. We also prove the p-parity conjecture for all elliptic curves over Q and all primes p: the parities of the p^\infty-Selmer rank and the analytic rank agree.

K3 surfaces with Picard rank 20

We determine all complex K3 surfaces with Picard rank 20 over ℚ. Here the Neron–Severi group has rank 20 and is generated by divisors which are defined over ℚ. Our proof uses modularity, the Artin–Tate conjecture and class group theory. With different techniques, the result has been established by Elkies to show that Mordell–Weil rank 18 over ℚ is impossible for an elliptic K3 surface. We apply our methods to general singular K3 surfaces, that is, those with Neron–Severi group of rank 20, but not necessarily generated by divisors over ℚ.