Mordell--Weil groups over large algebraic extensions of fields of characteristic zero
Takuya Asayama, Yuichiro Taguchi
备注：
17 pages, 1 table. The proofs of Theorems 5.2 and 5.4 in v1 contained errors. Addressing these issues has led to a weaker claim than the original version; see Theorem 5.3 in the revised version. Published in Res. Number Theory
We study the structure of the Mordell--Weil groups of semiabelian varieties over large algebraic extensions of a finitely generated field of characteristic zero. We consider two types of algebraic extensions in this paper; one is of extensions obtained by adjoining the coordinates of certain points of various semiabelian varieties; the other is of extensions obtained as the fixed subfield in an algebraically closed field by a finite number of automorphisms. Some of such fields turn out to be new examples of Kummer-faithful fields which are not sub-$p$-adic. Among them, we find both examples of Kummer-faithful fields over which the Mordell--Weil group modulo torsion can be free of infinite rank and not free.
In this paper we prove Hasse local-global principle for polynomials with coefficients in Mordell-Weil type groups over number fields like S-units, abelian varieties with trivial ring of endomorphisms and odd algebraic K-theory groups.
We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps, in the Mordell-Weil group of an abelian variety over a number field.
In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(ζ_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.
We present a method for computing the Mordell-Weil rank of the jacobian of a curve of genus 2 with multiplication by a square root of 2, based on descent via isogenies of degree 2, and apply it to a family of curves.
We develop a general classification theory for Brumer's dihedral quintic polynomials by means of Kummer theory arising from certain elliptic curves. We also give a similar theory for cubic polynomials.
In this paper, we examine certain maximal rank sublattices of the Mordell-Weil lattices of two families of elliptic curves over fields of characteristic $p > 0$. We compute explicit lower bounds on the densest sphere packings of these sublattices by finding lower bounds on the minimal norms of the sublattices and explicitly computing the volumes of their fundamental domains.
On the structure of fine Mordell-Weil groups over a $\mathbb{Z}_p$-extension and its intermediate subextensions
School of Mathematics and Statistics, Key Laboratory of Nonlinear Analysis and Applications (Ministry of Education), Central China Normal University（华中师范大学数学与统计学院，非线性分析与应用教育部重点实验室）
Answering a question of Ed Schaefer, we show that if J is the Jacobian of a curve C over a number field, if s is an automorphism of J coming from an automorphism of C, and if u lies in the subring Z[s] of End J and has connected kernel, then it is not necessarily the case that u gives a surjective map from the Mordell-Weil group of J to the Mordell-Weil group of its image.
We present an heuristic argument for the prediction of expected Mordell-Weil rank of elliptic curves over number fields, using Birch and Swinnerton-Dyer's original conjecture and Sato-Tate conjectures. We do calculations in some cases and raise questions about their relations, if any, with the predictions of various average rank models that have been considered.
We study an infinite family of Mordell curves (i.e. the elliptic curves in the form y^2=x^3+n, n \in Z) over Q with three explicit integral points. We show that the points are independent in certain cases. We describe how to compute bounds of the canonical heights of the points. Using the result we show that any pair in the three points can always be a part of a basis of the free part of the Mordell--Weil group.
If A is an abelian variety over a number field K, and L is a (possibly infinite) extension of K generated by torsion points of A, then the quotient of A(L) by its torsion subgroup is a free abelian group.
Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $φ-$Selmer groups, Shafarevich-Tate groups ($ φ- $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one are determined explicitly in many cases.
Let $K$ be a number field, $\bar{K}$ an algebraic closure of $K$ and $E/K$ an elliptic curve defined over $K$. In this paper, we prove that if $E/K$ has a $K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then for each $σ\in Gal(\bar{K}/K)$, the Mordell-Weil group $E(\bar{K}^σ)$ of $E$ over the fixed subfield of $\bar{K}$ under $σ$ has infinite rank.
Let $E/\bbq$ be an elliptic curve defined over $\bbq$ with conductor $N$ and $\gq$ the absolute Galois group of an algebraic closure $\bar{\bbq}$ of $\bbq$. We prove that for every $σ\in \gq$, the Mordell-Weil group $E(\oqs)$ of $E$ over the fixed subfield of $\bar{\bbq}$ under $σ$ has infinite rank. Our approach uses the modularity of $E/\bbq$ and a collection of algebraic points on $E$ -- the so-called {\em Heegner points} -- arising from the theory of complex multiplication.
We discuss the Mordell-Weil sieve as a general technique for proving results concerning rational points on a given curve. In the special case of curves of genus 2, we describe quite explicitly how the relevant local information can be obtained if one does not want to restrict to mod p information at primes of good reduction. We describe our implementation of the Mordell-Weil sieve algorithm and discuss its efficiency.
We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.
Let $p$ be an odd prime and $F_{\infty}$ a $p$-adic Lie extension of a number field $F$. Let $A$ be an abelian variety over $F$ which has ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell-Weil rank of the abelian variety of $A$ in the said $p$-adic Lie extension. Our upper bound can be expressed in terms of invariants coming from the cyclotomic level. Motivated by this formula, we make a conjecture on an asymptotic upper bound of the growth of Mordell-Weil ranks over a $p$-adic Lie extension which is in terms…
Bruin--Najman and Ozman--Siksek have recently determined the quadratic points on all modular curves $X_0(N)$ of genus 2, 3, 4, and 5 whose Mordell--Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell--Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67 and 73. The main tool used is a relative symmetric Chabauty method, in combination with the Mordell--Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this …
Let $F$ be a number field unramified at an odd prime $p$ and $F_\infty$ be the $\mathbf{Z}_p$-cyclotomic extension of $F$. Let $A$ be an abelian variety defined over $F$ with good supersingular reduction at all primes of $F$ above $p$. Büyükboduk and the first named author have defined modified Selmer groups associated to $A$ over $F_\infty$. Assuming that the Pontryagin dual of these Selmer groups are torsion $\mathbf{Z}_p[[\mathrm{Gal}(F_\infty/F)]]$-modules, we give an explicit sufficient condition for the rank of the Mordell-Weil group $A(F_n)$ to be bounded as $n$ varies.
Let $A$ be an abelian variety defined over a number field $K$. If $\mathfrak{p}$ is a prime of $K$ of good reduction for $A$, let $A(K)_\mathfrak{p}$ denote the image of the Mordell-Weil group via reduction modulo $\mathfrak{p}$. We prove in particular that the size of $A(K)_\mathfrak{p}$, by varying $\mathfrak{p}$, encodes enough information to determine the $K$-isogeny class of $A$, provided that the following necessary condition is satisfied: $B(K)$ has positive rank for every non-trivial abelian subvariety $B$ of $A$. This is the analogue to a result by Faltings of 1983 considering instead…
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.
This master thesis describes how Selmer groups can be used to determine the Mordell-Weil group of elliptic curves over a number field K. The Mordell-Weil Theorem states that $E(K) = E(K)_{tors} \times Z^r$, where $r$ is the rank of $E$, and $E(K)_{tors}$ is the torsion subgroup, i.e. the group of points of finite order in $E(K)$. The group $E(K)_{tors}$ is finite and well understood. So, one tries to find a way to determine the rank $r$ of $E$, which is the major problem. The procedure described in this thesis shows how to transfer the computation of the weak Mordell-Weil group $E(K)/mE(K)$ to…
In this paper we investigate linear dependence of points in Mordell-Weil groups of abelian varieties via reduction maps. In particular we try to determine the conditions for detecting linear dependence in Mordell-Weil groups via finite number of reductions.
In this article we extend work of Shanks and Washington on cyclic extensions, and elliptic curves associated to the simplest cubic fields. In particular, we give families of examples of hyperelliptic curves $C: y^2=f(x)$ defined over $\mathbb{Q}$, with $f(x)$ of degree $p$, where $p$ is a Sophie Germain prime, such that the rank of the Mordell--Weil group of the jacobian $J/\mathbb{Q}$ of $C$ is bounded by the genus of $C$ and the $2$-rank of the class group of the (cyclic) field defined by $f(x)$, and exhibit examples where this bound is sharp.
We prove that the elliptic surface y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1) has geometric Mordell-Weil rank 15. This completes a list of Kuwata, who gave explicit examples of elliptic K3-surfaces with geometric Mordell-Weil rank 0,1,..., 14, 16, 17,18.
In this paper we prove a special case of the Lehmer inequality for Drinfeld modules. Also, based on this inequality, we prove certain Mordell-Weil type of theorems for certain infinitely generated fields.
In this paper we establish a Hasse principle concerning the linear dependence over $\Z$ of nontorsion points in the Mordell-Weil group of an abelian variety over a number field.
We prove results on the Mordell--Weil rank of elliptic curves $y^2=x(x-αa^2)(x-βb^2)$ parametrized by binary quadratic forms $αa^2+βb^2=γc^2$. We express our explicit lower bounds over number fields and offer a detailed description of the corresponding Mordell-Weil group structure in the function field case.
We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and apply them with a 2-descent argument to obtain bounds for families of elliptic curves over certain imaginary quadratic number fields with class number 1. One such family occurs in the congruent number problem. We consider the congruent number problem over these quadratic number fields, and subject to the finiteness of Sha, we show that there are infinitely many numbers that are not congruent over Q but become congruent over $Q(ζ_3)$.
1) We present new lattice sphere packings in Euclid spaces of many dimensions in the range 3332-4096, which are denser than known densest Mrodell-Weil lattice sphere packings in these dimensions. Moreover it is proved that if there were some nice linear binary codes we could construct lattices denser than Mordell-Weil lattices of many dimensions in the range 116-3332. 2) New lattices with densities at least 8 times of the densities of Craig lattices in the dimensions $p-1$, where $p$ is a prime satisfying $p-1 \geq 1222$, are constructed. Some of these lattices provide new record sphere packin…
Let $E$ be an elliptic curve defined over a number field $K$ where $p$ splits completely. Suppose that $E$ has good reduction at all primes above $p$. Generalizing previous works of Kobayashi and Sprung, we define multiply signed Selmer groups over the cyclotomic $\mathbb{Z}_p$-extension of a finite extension $F$ of $K$ where $p$ is unramified. Under the hypothesis that the Pontryagin duals of these Selmer groups are torsion over the corresponding Iwasawa algebra, we show that the Mordell-Weil ranks of $E$ over a subextension of the cyclotomic $\mathbb{Z}_p$-extension are bounded. Furthermore,…
Let A be an abelian variety defined over a number field k and let F be a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we compute explicitly the algebraic part of the p-component of the equivariant Tamagawa number of the pair ( h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with the theorem of Gross and Zagier we are able to give the first verification of the p-component of the equivariant Tamagawa number conjecture for an abelian variety in the technically most demanding case in which the relevant Mor…
We study the variation of Mordell-Weil ranks in the Jacobians of curves in a pro-p tower over a fixed number field. In particular, we show that under mild conditions the Mordell-Weil rank of a Jacobian in the tower is bounded above by a constant multiple of its dimension. In the case of the tower of Fermat curves, we show that the constant can be taken arbitrarily close to 1. The main result is used in the forthcoming paper of Guillermo Mantilla-Soler on the Mordell-Weil rank of the modular Jacobian J(Np^m).
We consider an abelian variety defined over a number field. We give conditional bounds for the order of its Tate-Shafarevich group, as well as conditional bounds for the Néron-Tate height of generators of its Mordell-Weil group. The bounds are implied by strong but nowadays classical conjectures, such as the Birch and Swinnerton-Dyer conjecture and the functional equation of the L-series. In particular, we improve and generalise a result by D. Goldfeld and L. Szpiro on the order of the Tate-Shafarevich group, and extends a conjecture of S. Lang on the canonical height of a system of generators…
Let A be a modular abelian variety of GL2-type over a totally real field F of class number one. Under some mild assumptions, we show that the Mordell-Weil rank of A grows polynomially over Hilbert class fields of CM extensions of F.
We study the elliptic curves given by $y^2 = x^3 + b x + t^{3^n+1}$ over global function fields of characteristic $3$; in particular we perform an explicit computation of the $L$-function by relating it to the zeta function of a certain superelliptic curve $u^3 + b u = v^{3^n + 1}$. In this way, using the Néron-Tate height on the Mordell--Weil group, we obtain lattices in dimension $2 \cdot 3^n$ for every $n \geq 1$, which improve on the currently best known sphere packing densities in dimensions 162 (case $n=4$) and 486 (case $n=5$). For $n=3$, the construction has the same packing density as…
Let $E_{m,n}$ be an elliptic curve over $\mathbb{Q}$ of the form $y^2=x^3-m^2x+n^2$, where $m$ and $n$ are positive integers. Brown and Myers showed that the curve $E_{1,n}$ has rank at least two for all $n$. In the present paper, we specify the two points which can be extended to a basis for $E_{1,n}(\mathbb{Q})$ under certain conditions described explicitly. Moreover, we verify a similar result for the curve $E_{m,1}$, which, however, gives a basis for the rank three part of $E_{m,1}(\mathbb{Q})$.
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.
In the first paper of this series, we constructed a family of lattices in dimensions 2^{n+1} for positive integers n, and proved that the associated lattice packings of spheres equal or exceed the previous records for several values of n. In this paper we analyze the case n=6, which is now the first case where the resulting lattice MW128 yields a previously unknown lattice of record density. We prove that the elliptic curve used to construct this lattice has trivial Tate-Shafarevich group, and exhibit on that curve a rational point whose canonical height attains our lower bound on the minimal …
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 $k$ to the 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 $S$, for all vertical curves $S_x$ of a fibration $S \to U \subseteq \mathbf{P}^1_k$ with $x$ from the complement of a sparse subset of $|U|$, the Mordell-Weil rank of an abelian scheme o…
For a given number field $K$, we show that the ranks of nonsingular elliptic curves over $K$ are uniformly finitely bounded if and only if weak Mordell-Weil property holds in all(some) ultrpowers $^*K$ of $K$. Also we introduce Nonstandard Mordell-Weil property for $^*K$ considering each Mordell-Weil group as $^*Z$-module, where $^*Z$ is an ultrapower of $Z$, and we show that Nonstandard Mordell-Weil property is equivalent to weak Mordell-Weil property in $^*K$. In Appendix, we showed that it is possible to consider definable abelian groups as $^*Z$-modules in a saturated nonstandard rational …
We discuss the distribution of Mordell--Weil ranks of the family of elliptic curves $y^2=(x+αf^2)(x+βb g^2)(x+γh^2)$ where $f,g,h$ are coprime polynomials that parametrize the projective smooth conic $a^2+b^2=c^2$ and $α,β,γ$ are elements from $\overline{\mathbb{Q}}$. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
We use methods for computing Picard numbers of reductions of K3 surfaces in order to study the decomposability of Jacobians over number fields and the variance of Mordell-Weil ranks of families of Jacobians over different ground fields. For example, we give examples of surfaces whose Picard numbers jump in rank at all primes of good reduction using Mordell-Weil groups of Jacobians and show that the genus of curves over number fields whose Jacobians are isomorphic to a product of elliptic curves satisfying certain reduction conditions is bounded. The isomorphism result addresses the number fiel…
We study the arithmetic of abelian varieties over $K=k(t)$ where $k$ is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over $K$ to homomorphisms of other Jacobians over $k$. Our methods also yield completely explicit points on elliptic curves with unbounded rank over $\Fpbar(t)$ and a new construction of elliptic curves with moderately high rank over $\C(t)$.
Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve $C$, given only the $p$-Selmer group $S$ of its Jacobian (or some other abelian variety $C$ maps to) and the image of the $p$-Selmer set of $C$ in $S$. The method is more likely to succeed when the genus is large, which is when it is usually rather difficult to obtain generators of a finite-index subgroup of the Mordell-Weil group, which one would need to apply Chabauty's method in the usual way. We give some applications, for example to ge…
Consider elliptic curves $ E=E_σ: y^2 = x (x+σp) (x+σq), $ where$ σ=\pm 1, $ $p$ and $ q$ are prime numbers with $p+2=q$. (1) The Selmer groups $ S^{(2)}(E/{\mathbf{Q}}), S^{(ϕ)}(E/{\mathbf{Q})}$, and $\ S^{(\hatϕ)}(E/{\mathbf{Q})} $ are explicitly determined, e.g., $\ S^{(2)}(E_{+1}/{\mathbf{Q}})= $ $({\mathbf{Z}}/2{\mathbf{Z}})^2; $ $ ({\mathbf{Z}}/2{\mathbf{Z}})^3; $ or $ ({\mathbf{Z}}/2{\mathbf{Z}})^4 $ when $p\equiv 5; 1 $ or $3; $ or $ 7 ({\mathrm{mod}} 8)$ respectively. (2) When $p\equiv 5 (3, 5$ for $σ=-1) ({\mathrm{mod}} 8), $ it is proved that the Mordell-Weil group $ E({\mathbf{Q})}…
We give a completely explicit upper bound for integral points on (standard) affine models of hyperelliptic curves, provided we know at least one rational point and a Mordell-Weil basis of the Jacobian. We also explain a powerful refinement of the Mordell--Weil sieve which, combined with the upper bound, is capable of determining all the integral points. Our method is illustrated by determining the integral points on a two genus 2 hyperelliptic curves with Mordell--Weil Jacobian ranks of 3 and 6.
Let $E$ be an elliptic curve defined over a number field $F$ with good ordinary reduction at all primes above $p$, and let $F_\infty$ be a finitely ramified uniform pro-$p$ extension of $F$ containing the cyclotomic $\mathbb{Z}_p$-extension $F_{cyc}$. Set $F^{(n)}$ be the $n$-th layer of the tower, and $F^{(n)}_{cyc}$ the cyclotomic $\mathbb{Z}_p$-extension of $F^{(n)}$. We study the growth of the rank of $E(F^{(n)})$ by analyzing the growth of the $λ$-invariant of the Selmer group over $F^{(n)}_{cyc}$ as $n\rightarrow \infty$. This method has its origins in work of A.Cuoco, who studied $\math…
