Torsion Points Research Articles

Galois orbits of torsion points near atoral sets

Galois orbits of torsion points near atoral sets

On a Galois property of fields generated by the torsion of an abelian variety

AbstractIn this article, we study a certain Galois property of subextensions of , the minimal field of definition of all torsion points of an abelian variety defined over a number field . Concretely, we show that each subfield of that is Galois over (of possibly infinite degree) and whose Galois group has finite exponent is contained in an abelian extension of some finite extension of . As an immediate corollary of this result and a theorem of Bombieri and Zannier, we deduce that each such field has the Northcott property, that is, does not contain any infinite set of algebraic numbers of bounded height.

Two-torsion subgroups of some modular Jacobians

We give a practical method to compute the 2-torsion subgroup of the Jacobian of a non-hyperelliptic curve of genus [Formula: see text], [Formula: see text] or [Formula: see text]. The method is based on the correspondence between the 2-torsion subgroup and the theta hyperplanes to the curve. The correspondence is used to explicitly write down a zero-dimensional scheme whose points correspond to elements of the [Formula: see text]-torsion subgroup. Using [Formula: see text]-adic or complex approximations (obtained via Hensel lifting or homotopy continuation and Newton–Raphson) and lattice reduction we are then able to determine the points of our zero-dimensional scheme and hence the [Formula: see text]-torsion points. We demonstrate the practicality of our method by computing the [Formula: see text]-torsion of the modular Jacobians [Formula: see text] for [Formula: see text]. As a result of this we are able to verify the generalized Ogg conjecture for these values.

Elliptic curves with a rational 2-torsion point ordered by conductor and the boundedness of average rank

In this paper we refine recent work due to A. Shankar, A. N. Shankar, and X. Wang on counting elliptic curves by conductor to the case of elliptic curves with a rational 2-torsion point. This family is a small family, as opposed to the large families considered by the aforementioned authors. We prove the analogous counting theorem for elliptic curves with so-called square-free index as well as for curves with suitably bounded Szpiro ratios. We note that the assumptions we impose on the size of the Szpiro ratios are less stringent than would be expected by the naive generalization of their approach; this requires an application of a theorem of Browning and Heath-Brown. Our proof also relies on linear programming techniques.

Torsion points of elliptic curves over multi-quadratic number fields

We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X1(M,MN) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.

Efficient computation of (2n,2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2^n,2^n)$$\\end{document}-isogenies

Elliptic curves are abelian varieties of dimension one; the two-dimensional analogues are abelian surfaces. In this work we present an algorithm to compute (2n,2n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(2^n,2^n)$$\\end{document}-isogenies between abelian surfaces defined over finite fields. These isogenies are the natural generalization of 2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^n$$\\end{document}-isogenies of elliptic curves. The efficient computation of such isogeny chains gained a lot of attention as the runtime of the attacks on SIDH (Castryck–Decru, Maino–Martindale, Robert) depends on this computation. Different results deduced in the development of our algorithm are also interesting beyond these applications. For instance, we derive a formula for the evaluation of (2, 2)-isogenies. Given an element in Mumford coordinates, this formula outputs the (unreduced) Mumford coordinates of its image under the (2, 2)-isogeny. Furthermore, we study 4-torsion points on Jacobians of hyperelliptic curves and explain how to extract square roots of coefficients of 2-torsion points from these points.

A signed count of 2-torsion points on real abelian varieties

A signed count of 2-torsion points on real abelian varieties

INTERSECTING THE TORSION OF ELLIPTIC CURVES

Abstract Bogomolov and Tschinkel [‘Algebraic varieties over small fields’, Diophantine Geometry, U. Zannier (ed.), CRM Series, 4 (Scuola Normale Superiore di Pisa, Pisa, 2007), 73–91] proved that, given two complex elliptic curves $E_1$ and $E_2$ along with even degree- $2$ maps $\pi _j\colon E_j\to \mathbb {P}^1$ having different branch loci, the intersection of the image of the torsion points of $E_1$ and $E_2$ under their respective $\pi _j$ is finite. They conjectured (also in works with Fu) that the cardinality of this intersection is uniformly bounded independently of the elliptic curves. The recent proof of the uniform Manin–Mumford conjecture implies a full solution of the Bogomolov–Fu–Tschinkel conjecture. In this paper, we prove a generalisation of the Bogomolov–Fu–Tschinkel conjecture whereby, instead of even degree- $2$ maps, one can use any rational functions of bounded degree on the elliptic curves as long as they have different branch loci. Our approach combines Nevanlinna theory with the uniform Manin–Mumford conjecture. With similar techniques, we also prove a result on lower bounds for ranks of elliptic curves over number fields.

Local–global divisibility on algebraic tori

AbstractWe give a complete answer to the local–global divisibility problem for algebraic tori. In particular, we prove that given an odd prime , if is an algebraic torus of dimension defined over a number field , then the local–global divisibility by any power holds for . We also show that this bound on the dimension is best possible, by providing a counterexample for every dimension . Finally, we prove that under certain hypotheses on the number field generated by the coordinates of the ‐torsion points of , the local–global divisibility still holds for tori of dimension less than .

Pro-$p$ Criterion of Good Reduction of Punctured Elliptic Curves

In this paper we consider an affine hyperbolic curve X over a p -adic local field K obtained from an elliptic curve E by removing a K -rational point given by the origin O . We introduce the maximal geometrically pro- p quotient \Pi_X of the étale fundamental group \pi_1^{\mathrm{ét}}(X) of the hyperbolic curve X and analyse the problem of determining the reduction type of the elliptic curve E from the topological group \Pi_X . As our main result we give a group-theoretic construction of the reduction type of E from the group \Pi_X under the assumption that p\ge5 and that E has a nontrivial K -rational p -torsion point.

Ramified covers of abelian varieties over torsion fields

Abstract We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of ℚ {\mathbb{Q}} . In particular, we prove that every elliptic curve E over ℚ {\mathbb{Q}} has the weak Hilbert property of Corvaja and Zannier both over the maximal abelian extension ℚ ab {\mathbb{Q}^{\rm ab}} of ℚ {\mathbb{Q}} , and over the field ℚ ⁢ ( A tor ) {\mathbb{Q}(A_{\rm tor})} obtained by adjoining to ℚ {\mathbb{Q}} all torsion points of some abelian variety A over ℚ {\mathbb{Q}} .

ON THE ESSENTIAL TORSION FINITENESS OF ABELIAN VARIETIES OVER TORSION FIELDS

AbstractThe classical Mordell–Weil theorem implies that an abelian variety A over a number field K has only finitely many K-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm {cyc}}=K{\mathbb Q}^{\mathrm {ab}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety A over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety B. Assuming the Mumford–Tate conjecture, and up to a finite extension of the base field K, we give a necessary and sufficient condition for the finiteness of $A(K_B)_{\mathrm tors}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension at most 3, or when both have complex multiplication.

The Tropical Manin–Mumford Conjecture

Abstract In analogy with the Manin–Mumford conjecture for algebraic curves, one may ask how a metric graph under the Abel–Jacobi embedding intersects torsion points of its Jacobian. We show that the number of torsion points is finite for metric graphs of genus ${g\geq 2}$, which are biconnected and have edge lengths that are “sufficiently irrational” in a precise sense. Under these assumptions, the number of torsion points is bounded by $3g-3$. Next, we study bounds on the number of torsion points in the image of higher-degree Abel–Jacobi embeddings, which send $d$-tuples of points to the Jacobian. This motivates the definition of the “independent girth” of a graph, a number that is a sharp upper bound for $d$ such that the higher-degree Manin–Mumford property holds.

Coincidences of division fields

Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $\rho_E\colon {\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to {\rm GL}(2,\widehat{ \mathbb{Z} })$ be the adelic representation associated to the natural action of Galois on the torsion points of $E(\overline{\mathbb{Q}})$. By a theorem of Serre, the image of $\rho_{E}$ is open, but the image is always of index at least $2$ in ${\rm GL}(2,\widehat{\mathbb{Z}})$ due to a certain quadratic entanglement amongst division fields. In this paper, we study other types of abelian entanglements. More concretely, we classify the elliptic curves $E/\mathbb{Q}$, and primes $p$ and $q$ such that $\mathbb{Q}(E[p])\cap \mathbb{Q}(\zeta_{q^k})$ is non-trivial, and determine the degree of the coincidence. As a consequence, we classify all elliptic curves $E/\mathbb{Q}$ and integers $m,n$ such that the $m$-th and $n$-th division fields coincide, i.e., when $\mathbb{Q}(E[n])=\mathbb{Q}(E[m])$, when the division field is abelian.

Ramification of p-power torsion points of formal groups

Let p be a rational prime, let F denote a finite, unramified extension of $$\mathbb {Q}_p$$ , let K be the completion of the maximal unramified extension of $$\mathbb {Q}_p$$ , and let $$\overline{K}$$ be some fixed algebraic closure of K. Let A be an abelian variety defined over F, with good reduction, let $$\mathcal {A}$$ denote the Néron model of A over $$\textrm{Spec}(\mathcal {O}_F)$$ , and let $$\widehat{\mathcal {A}}$$ be the formal completion of $$\mathcal {A}$$ along the identity of its special fiber, i.e. the formal group of A. In this work, we prove two results concerning the ramification of p-power torsion points on $$\widehat{\mathcal {A}}$$ . One of our main results describes conditions on $$\widehat{\mathcal {A}}$$ , base changed to $$\text {Spf}(\mathcal {O}_K) $$ , for which the field $$K(\widehat{\mathcal {A}}[p])/K$$ i s a tamely ramified extension where $$\widehat{\mathcal {A}}[p]$$ denotes the group of p-torsion points of $$\widehat{\mathcal {A}}$$ over $$\mathcal {O}_{\overline{K}}$$ . This result generalizes previous work when A is 1-dimensional and work of Arias-de-Reyna when A is the Jacobian of certain genus 2 hyperelliptic curves.

Canonical equivariant cohomology classes generating zeta values of totally real fields

It is known that the special values at nonpositive integers of a Dirichlet L L -function may be expressed using the generalized Bernoulli numbers, which are defined by a generating function. The purpose of this article is to consider the generalization of this classical result to the case of Hecke L L -functions of totally real fields. Hecke L L -functions may be expressed canonically as a finite sum of zeta functions of Lerch type. By combining the non-canonical multivariable generating functions constructed by Shintani [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), pp. 393–417], we newly construct a canonical class, which we call the Shintani generating class, in the equivariant cohomology of an algebraic torus associated to the totally real field. Our main result states that the specializations at torsion points of the derivatives of the Shintani generating class give values at nonpositive integers of the zeta functions of Lerch type. This result gives the insight that the correct framework in the higher dimensional case is to consider higher equivariant cohomology classes instead of functions.

Torsion of algebraic groups and iterate extensions associated with Lubin–Tate formal groups

We show finiteness results on torsion points of commutative algebraic groups over a $p$-adic field $K$ with values in various algebraic extensions $L/K$ of infinite degree. We mainly study the following cases: (1) $L$ is an abelian extension which is a splitting field of a crystalline character (such as a Lubin–Tate extension). (2) $L$ is a certain iterate extension of $K$ associated with points on Lubin–Tate formal groups.

Strain-Insensitive Simultaneous Measurements of Torsion and Temperature Using Long-Period Fiber Grating Inscribed on Twisted Double-Clad Fiber

Here, we report an optical fiber sensor capable of performing strain-insensitive simultaneous measurements of torsion and temperature using a long-period fiber grating (LPFG) inscribed on twisted double-clad fiber (DCF) with a CO <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> laser. The LPFG written on twisted DCF, referred to as TDC-LPFG, exhibited multiple resonance dips with different cladding-mode orders, and two of them were selected as sensor indicators for the simultaneous measurement of torsion and temperature. The measured torsion sensitivities of the sensor indicators were approximately –2.167×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> and –4.667×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> pm/(rad/m) in a twist angle range of –180° to 180°, and their temperature sensitivities were measured as ~6.580×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> and ~7.189×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> pm/°C in a temperature range of 25–100 °C. Their linear and independent responses to torsion and temperature made possible the discrimination of the two measurands. Moreover, the strain sensitivity of the TDC-LPFG was measured and found to be less than 2.750×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> pm/με in magnitude, over four times less than that of a standard LPFG. This strain insensitiveness contributes strongly to diminishing strain-induced measurement uncertainties associated with torsion and temperature. Finally, we validated the simultaneous measurement capability of twenty-five applied points of torsion and temperature variations. The average torsion and temperature standard deviations were ~3.44×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> rad/m and ~3.01×10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> °C, respectively.

Torsion points on elliptic curves over number fields of small degree

We determine the set $S(d)$ of possible prime orders of $K$-rational points on elliptic curves over number fields $K$ of degree $d$, for $d = 4$, $5$, $6$, and $7$.

Elliptic R-matrices and Feigin and Odesskii’s elliptic algebras

The algebras $$Q_{n,k}(E,\tau )$$ introduced by Feigin and Odesskii as generalizations of the 4-dimensional Sklyanin algebras form a family of quadratic algebras parametrized by coprime integers $$n>k\ge 1$$ , a complex elliptic curve E, and a point $$\tau \in E$$ . The main result in this paper is that $$Q_{n,k}(E,\tau )$$ has the same Hilbert series as the polynomial ring on n variables when $$\tau $$ is not a torsion point. We also show that $$Q_{n,k}(E,\tau )$$ is a Koszul algebra, hence of global dimension n when $$\tau $$ is not a torsion point, and, for all but countably many $$\tau $$ , $$Q_{n,k}(E,\tau )$$ is Artin–Schelter regular. The proofs use the fact that the space of quadratic relations defining $$Q_{n,k}(E,\tau )$$ is the image of an operator $$R_{\tau }(\tau )$$ that belongs to a family of operators $$R_{\tau }(z):{\mathbb {C}}^n\otimes {\mathbb {C}}^n\rightarrow {\mathbb {C}}^n\otimes {\mathbb {C}}^n$$ , $$z\in {\mathbb {C}}$$ , that (we will show) satisfy the quantum Yang–Baxter equation with spectral parameter.