On the modularity of elliptic curves over -adic exercises

We complete the proof that every elliptic curve over the rational numbers is modular.

     

On construction of certain CM elliptic curves

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268-286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ₁(N) for some N such that HomF(E,Jf)≠{0}. We also determined the multiplicity of E as F-simple factor of Jf when HomF(E,Jf)≠{0}. In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Modularity of CM elliptic curves over division fields

Let E be a CM elliptic curve defined over an algebraic number field F. In general E will not be modular over F. In this paper, we determine extensions of F, contained in suitable division fields of E, over which E is modular. Under some weak assumptions on E, we construct a minimal subfield of division fields over which E is modular.     

On the Tate-Shafarevich groups of certain elliptic curves

The Tate-Shafarevich groups of certain elliptic curves over Fq(t) are related, via étale cohomology, to the group of points of an elliptic curve with complex multiplication. The Cassels-Tate pairing is computed under this identification.     

Construction of high-rank elliptic curves with a nontrivial torsion point

We construct a family of infinitely many elliptic curves over with a nontrivial rational 2-torsion point and with rank 6, which is parametrized by the rational points of an elliptic curve of rank 1.

     

On the discrete logarithm problem for prime-field elliptic curves

In recent years several papers have appeared that investigate the classical discrete logarithm problem for elliptic curves by means of the multivariate polynomial approach based on the celebrated summation polynomials, introduced by Semaev in 2004. With a notable exception by Petit et al. in 2016, all numerous papers on the subject have investigated only the composite-field case, leaving apart the laborious prime-field case. In this paper we propose a variation of Semaev's original approach that reduces to only one the relations to be found among points of the factor base, thus decreasing drastically the necessary Groebner basis computations. Our proposal holds for any finite field but it is particularly suitable for the prime-field case, where it outperforms both the original Semaev's method and the specialised algorithm by Petit et al..     

On the size of the Shafarevich--Tate group of elliptic curves over function fields

Let E be a nonconstant elliptic curve, over a global field K of positive, odd characterisitc. Assuming the finiteness of the Shafarevich-Tate group of E, we show that the order of theShafarevich-Tate group of E, is given by O ^{(N1/2+6 log(2)/ log(q))}, where N is the conductor of E,q isthe cardinality of the finite field of constants of K, and where the constant in the bound depends only on K. The method of proof is to workwith the geometric analog of the Birch-Swinnerton Dyer conjecture for thecorresponding elliptic surface over the finite field, as formulatedby Artin-Tate, and to examine the geometry of this elliptic surface.     

Torsion on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .

     

Selmer groups of elliptic curves that can be arbitrarily large

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X₀(p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p?7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.     

Rings of algebraic numbers in infinite extensions of $${\mathbb {Q}}$$ and elliptic curves retaining their rank

We show that elliptic curves whose Mordell–Weil groups are finitely generated over some infinite extensions of , can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.      

The density of elliptic curves having a global minimal Weierstrass equation

We show that a positive density of elliptic curves over a number field counted using their short Weierstrass equations belong to a given Weierstrass class and in particular, a positive density of elliptic curves have a global minimal Weierstrass equation. The density is given by a ratio of partial zeta functions of the number field K evaluated at 10 with some extra factors for the bad primes.     

On standardized models of isogenous elliptic curves

Let be isogenous elliptic curves over given by standardized Weierstrass models. We show that (in the obvious notation)

and, moreover, that there are integers such that

where .

     

On the field of definition of p-torsion points on elliptic curves over the rationals

Let $S_\mathbb Q (d)$ be the set of primes $p$ for which there exists a number field $K$ of degree $\le d$ and an elliptic curve $E/\mathbb Q $ , such that the order of the torsion subgroup of $E(K)$ is divisible by $p$ . In this article we give bounds for the primes in the set $S_\mathbb Q (d)$ . In particular, we show that, if $p\ge 11$ , $p\ne 13,37$ , and $p\in S_\mathbb Q (d)$ , then $p\le 2d+1$ . Moreover, we determine $S_\mathbb Q (d)$ for all $d\le 42$ , and give a conjectural formula for all $d\ge 1$ . If Serre’s uniformity problem is answered positively, then our conjectural formula is valid for all sufficiently large $d$ . Under further assumptions on the non-cuspidal points on modular curves that parametrize those $j$ -invariants associated to Cartan subgroups, the formula is valid for all $d\ge 1$ .     

Stark conjectures for CM elliptic curves over number fields

We present an elliptic curve analog of the Stark conjecture for the value of the L-function at s=0. Although implied by the general Beilinson conjectures, the approach here is very concrete. Several cases are proved.     

Addition law structure of elliptic curves

The study of alternative models for elliptic curves has found recent interest from cryptographic applications, after it was recognized that such models provide more efficiently computable algorithms for the group law than the standard Weierstrass model. Examples of such models arise via symmetries induced by a rational torsion structure. We analyze the module structure of the space of sections of the addition morphisms, determine explicit dimension formulas for the spaces of sections and their eigenspaces under the action of torsion groups, and apply this to specific models of elliptic curves with parametrized torsion subgroups.     

Perfect squares representing the number of rational points on elliptic curves over finite field extensions

Let q be a perfect power of a prime number p and $E({\mathbb{F}}_q)$ be an elliptic curve over ${\mathbb{F}}_q$ given by the equation $y^2=x^3+Ax+B$. For a positive integer n we denote by $\mathrm{\#}E({\mathbb{F}}_{q^n})$ the number of rational points on E (including infinity) over the extension ${\mathbb{F}}_{q^n}$. Under a mild technical condition, we show that the sequence ${\{\mathrm{\#}E({\mathbb{F}}_{q^n})\}}_{n>0}$ contains at most 10²⁰⁰ perfect squares. If the mild condition is not satisfied, then $\mathrm{\#}E({\mathbb{F}}_{q^n})$ is a perfect square for infinitely many n including all the multiples of 12. Our proof uses a quantitative version of the Subspace Theorem. We also find all the perfect squares for all such sequences in the range $q<50$ and $n\le 1000$.     

On the modularity of supersingular elliptic curves over certain totally real number fields

We study generalisations to totally real fields of the methods originating with Wiles and Taylor and Wiles [A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. 141 (1995) 443-551; R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. 141 (1995) 553-572]. In view of the results of Skinner and Wiles [C. Skinner, A. Wiles, Nearly ordinary deformations of irreducible residual representations, Ann. Fac. Sci. Toulouse Math. (6) 10 (2001) 185-215] on elliptic curves with ordinary reduction, we focus here on the case of supersingular reduction. Combining these, we then obtain some partial results on the modularity problem for semistable elliptic curves, and end by giving some applications of our results, for example proving the modularity of all semistable elliptic curves over .