Torsion bounds for elliptic curves and Drinfeld modules

We derive upper bounds on the number of L-rational torsion points on a given elliptic curve or Drinfeld module defined over a finitely generated field K, as a function of the degree [L:K]. Our main tool is the adelic openness of the image of Galois representations, due to Serre, Pink and Rütsche. Our approach is to prove a general result for certain Galois modules, which applies simultaneously to elliptic curves and to Drinfeld modules.     

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.     

Visualizing elements of order two in the Weil-Châtelet group

Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σ∈H¹(G_K,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.     

Rational points of bounded height on projective surfaces

We give upper bounds for the number of rational points of bounded height on the complement of the lines on projective surfaces.     

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 .     

Primitive divisors of elliptic divisibility sequences

Silverman proved the analogue of Zsigmondy's Theorem for elliptic divisibility sequences. For elliptic curves in global minimal form, it seems likely this result is true in a uniform manner. We present such a result for certain infinite families of curves and points. Our methods allow the first explicit examples of the elliptic Zsigmondy Theorem to be exhibited. As an application, we show that every term beyond the fourth of the Somos-4 sequence has a primitive divisor.     

Explicit Heegner points: Kolyvagin's conjecture and non-trivial elements in the Shafarevich-Tate group

Kolyvagin used Heegner points to associate a system of cohomology classes to an elliptic curve over Q and conjectured that the system contains a non-trivial class. His conjecture has profound implications on the structure of Selmer groups. We provide new computational and theoretical evidence for Kolyvagin's conjecture. More precisely, we explicitly approximate Heegner points over ring class fields and use these points to give evidence for the conjecture for specific elliptic curves of rank two. We explain how Kolyvagin's conjecture implies that if the analytic rank of an elliptic curve is at least two then the Zp-corank of the corresponding Selmer group is at least two as well. We also use explicitly computed Heegner points to produce non-trivial classes in the Shafarevich-Tate group.     

Height difference bounds for elliptic curves over number fields

Let E be an elliptic curve over a number field K. Let h be the logarithmic (or Weil) height on E and be the canonical height on E. Bounds for the difference are of tremendous theoretical and practical importance. It is possible to decompose as a weighted sum of continuous bounded functions Ψ_υ:E(K_υ)→R over the set of places υ of K. A standard method for bounding , (due to Lang, and previously employed by Silverman) is to bound each function Ψ_υ and sum these local ‘contributions’.In this paper, we give simple formulae for the extreme values of Ψ_υ for non-archimedean υ in terms of the Tamagawa index and Kodaira symbol of the curve at υ.For real archimedean υ a method for sharply bounding Ψ_υ was previously given by Siksek [Rocky Mountain J. Math. 25(4) (1990) 1501]. We complement this by giving two methods for sharply bounding Ψ_υ for complex archimedean υ.     

Cohomological Mackey functors in number theory

We prove relations between the evaluations of cohomological Mackey functors over complete discrete valuation rings or fields and apply this to Mackey functors that arise naturally in number theory. This provides relations between λ- and μ-invariants in Iwasawa theory, between Mordell-Weil groups, Shafarevich-Tate groups, Selmer groups and zeta functions of elliptic curves, and between ideal class groups and regulators of number fields.     

A family of quintic cyclic fields with even class number parameterized by rational points on an elliptic curve

We give a family of quintic cyclic fields with even class number parametrized by rational points on an elliptic curve associated with Emma Lehmer's quintic polynomial. Further, we use the arithmetic of elliptic curves and the Chebotarev density theorem to show that there are infinitely many such fields.     

K3 surfaces, rational curves, and rational points

We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.     

Proof of an exceptional zero conjecture for elliptic curves over function fields

Based on the analogy between number fields and function fields of one variable over finite fields, we formulate and prove an analogue of the exceptional zero conjecture of Mazur, Tate and Teitelbaum for elliptic curves defined over function fields. The proof uses modular parametrization by Drinfeld modular curves and the theory of non-archimedean integration. As an application we prove a refinement of the Birch-Swinnerton-Dyer conjecture if the analytic rank of the elliptic curve is zero.     

Complete classification of torsion of elliptic curves over quadratic cyclotomic fields

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In a previous paper Najman (in press) [9], the author examined the possible torsions of an elliptic curve over the quadratic fields Q(i) and . Although all the possible torsions were found if the elliptic curve has rational coefficients, we were unable to eliminate some possibilities for the torsion if the elliptic curve has coefficients that are not rational. In this note, by finding all the points of two hyperelliptic curves over Q(i) and , we solve this problem completely and thus obtain a classification of all possible torsions of elliptic curves over Q(i) and .

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=VPhCkJTGB_o.     

On the tensor rank of the multiplication in the finite fields

First, we prove the existence of certain types of non-special divisors of degree g−1 in the algebraic function fields of genus g defined over Fq. Then, it enables us to obtain upper bounds of the tensor rank of the multiplication in any extension of quadratic finite fields Fq by using Shimura and modular curves defined over Fq. From the preceding results, we obtain upper bounds of the tensor rank of the multiplication in any extension of certain non-quadratic finite fields Fq, notably in the case of F₂. These upper bounds attain the best asymptotic upper bounds of Shparlinski-Tsfasman-Vladut [I.E. Shparlinski, M.A. Tsfasman, S.G. Vladut, Curves with many points and multiplication in finite fields, in: Lecture Notes in Math., vol. 1518, Springer-Verlag, Berlin, 1992, pp. 145-169].     

A lower bound for the canonical height on elliptic curves over abelian extensions

Let E/K be an elliptic curve defined over a number field, let ? be the canonical height on E, and let K^ab/K be the maximal abelian extension of K. Extending work of M. Baker (IMRN 29 (2003) 1571-1582), we prove that there is a constant C(E/K)>0 so that every nontorsion point P∈E(K^ab) satisfies .     

Elliptic units in K2

We present certain norm-compatible systems in K₂ of function fields of some CM elliptic curves. We demonstrate that these systems have some properties similar to elliptic units.     

Constructing one-parameter families of elliptic curves with moderate rank

We give several new constructions for moderate rank elliptic curves over Q(T). In particular we construct infinitely many rational elliptic surfaces (not in Weierstrass form) of rank 6 over Q using polynomials of degree two in T. While our method generates linearly independent points, we are able to show the rank is exactly 6 without having to verify the points are independent. The method generalizes; however, the higher rank surfaces are not rational, and we need to check that the constructed points are linearly independent.     

Effective equidistribution and the Sato-Tate law for families of elliptic curves

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Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate law. We present two methods of proof. Both use the framework of Murty and Sinha (2009) [MS]; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=faW2iDpe5IE.     

Bernoulli-Hurwitz numbers, Wieferich primes and Galois representations

Let K be a quadratic imaginary number field with discriminant DK≠−3,−4 and class number one. Fix a prime p?7 which is unramified in K. Given an elliptic curve A/Q with complex multiplication by K, let be the representation which arises from the action of Galois on the Tate module. Herein it is shown that, for all but finitely many inert primes p, the image of a certain deformation of is “as large as possible”, that is, it is the full inverse image of a Cartan subgroup of SL(2,Zp). If p splits in K, then the same result holds as long as a certain Bernoulli-Hurwitz number is a p-adic unit which, in turn, is equivalent to a prime ideal not being a Wieferich place. The proof rests on the theory of elliptic units of Robert and Kubert-Lang, and on the two-variable main conjecture of Iwasawa theory for quadratic imaginary fields.