Visibility of ideal classes

Cremona, Mazur, and others have studied what they call visibility of elements of Shafarevich–Tate groups of elliptic curves. The analogue for an abelian number field K is capitulation of ideal classes of K in the minimal cyclotomic field containing K. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of non-trivial Shafarevich–Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0.     

Potential modularity for elliptic curves and some applications

In this paper we prove the simultaneous potential modularity for a finite number of elliptic curves defined over a totally real field. As an application we prove the meromorphic continuation of some L-functions associated to elliptic curves and Tate conjecture for a product of 2 or 4 elliptic curves defined over a totally real field.     

关于某些椭圆曲线的Tate—Shafarevich群的注记

本文首先考虑某个四元代数，通过对此四元代数的算术的研究，得到某类丢番图方程的解数与某些虚二次域类数之间的关系，最后，应用Ｔｕｎｎｅｌｌ的定理，得到一簇椭圆曲线的Ｔａｔｅ－Ｓｈａｆａｒｅｖｉｃｈ群的阶的上界。     

Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton–Dyer conjecture

Let E be an elliptic curve over Q of conductor N and K be an imaginary quadratic field, where all prime divisors of N split. If the analytic rank of E over K is equal to 1, then the Gross and Zagier formula for the value of the derivative of the L-function of E over K, when combined with the Birch and Swinnerton–Dyer conjecture, gives a conjectural formula for the order of the Shafarevich–Tate group of E over K. In this paper, we show that there are infinitely many elliptic curves E such that for a positive proportion of imaginary quadratic fields K, the 3-part of the conjectural formula is true.     

Low-cost addition–subtraction sequences for the final exponentiation in pairings

In this paper, we address the problem of finding low cost addition–subtraction sequences for situations where a doubling step is significantly cheaper than a non-doubling one. One application of this setting appears in the computation of the final exponentiation step of the reduced Tate pairing defined on ordinary elliptic curves. In particular, we report efficient addition–subtraction sequences for the Kachisa–Schaefer–Scott family of pairing-friendly elliptic curves, whose parameters involve computing the multi-exponentiation of relatively large sequences of exponents with a size of up to 26 bits.     

Non-vanishing of central L-values of canonical CM elliptic curves with quadratic twists

The aim of this paper is to extend results of Rorlich, Villegas and Yang about the non-vanishing of central L-values of canonical characters of imaginary quadratic fields over the rationals. One of the new ingredients in our paper is the local computations at the place “2”. Therefore, we extend their non-vanishing results to include imaginary quadratic fields of even discriminant. As a consequence, we show that the rank of the Mordell–Weil groups of certain canonical CM elliptic curves are zero.     

On a generalization of Kronecker’s limit formula

In this paper, we first generalize the Kronecker limit formula for a class of Epstein zeta functions using new approximation formulas. This enables us to derive some applications to the class number of quadratic imaginary number fields K and the period ratios of elliptic curves with complex multiplication.     

The Sato–Tate distribution in families of elliptic curves with a rational parameter of bounded height

We obtain new results concerning the Sato–Tate conjecture on the distribution of Frobenius angles over parametric families of elliptic curves with a rational parameter of bounded height.     

High rank elliptic curves with prescribed torsion group over quadratic fields

There are 26 possibilities for the torsion groups of elliptic curves defined over quadratic number fields. We present examples of high rank elliptic curves with a given torsion group which set the current rank records for most of the torsion groups. In particular, we show that for each possible torsion group, except maybe for \(\mathbb {Z}/15\mathbb {Z}\) , there exists an elliptic curve over some quadratic field with this torsion group and with rank \(\ge 2\) .     

Cuspidal representations in the cohomology of Deligne–Lusztig varieties for GL(2) over finite rings

We define closed subvarieties of some Deligne–Lusztig varieties for GL(2) over finite rings and study their ´etale cohomology. As a result, we show that cuspidal representations appear in it. Such closed varieties are studied in [Lus2] in a special case. We can do the same things for a Deligne–Lusztig variety associated to a quaternion division algebra over a non-archimedean local field. A product of such varieties can be regarded as an affine bundle over a curve. The base curve appears as an open subscheme of a union of irreducible components of the stable reduction of the Lubin–Tate curve in a special case. Finally, we state some conjecture on a part of the stable reduction using the above varieties. This is an attempt to understand bad reduction of Lubin–Tate curves via Deligne–Lusztig varieties.     

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 p-Descent for Jacobians of Hermitian Curves

Let X be the Fermat curve of degree q+1 over the field k of q² elements, where q is some prime power. Considering the Jacobian J of X as a constant abelian variety over the function field k(X), we calculate the multiplicities, in subfactors of the Shafarevich–Tate group, of representations associated with the action on X of a finite unitary group. J is isogenous to a power of a supersingular elliptic curve E, the structure of whose Shafarevich–Tate group is also described.     

Special automorphisms on Shimura curves and non-triviality of Heegner points

We define the notion of special automorphisms on Shimura curves. Using this notion, for a wild class of elliptic curves defined over Q, we get rank one quadratic twists by discriminants having any prescribed number of prime factors. Finally, as an application, we obtain some new results on Birch and Swinnerton-Dyer (BSD) conjecture for the rank one quadratic twists of the elliptic curve X₀(49).     

Dualité pour les groupes de type multiplicatif sur certains corps de fonctions

Let K be the function field of a smooth projective curve X over a higher-dimensional local field k. We define Tate–Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of K coming from a closed point of X. In this note, we state and sketch the proof of an arithmetic duality theorem for Tate–Shafarevich groups of groups of multiplicative type over K (and more generally of some two-term complexes of tori over K).     

Faster computation of the Tate pairing

Text

This paper proposes new explicit formulas for the doubling and addition steps in Miller's algorithm to compute the Tate pairing on elliptic curves in Weierstrass and in Edwards form. For Edwards curves the formulas come from a new way of seeing the arithmetic. We state the first geometric interpretation of the group law on Edwards curves by presenting the functions which arise in addition and doubling. The Tate pairing on Edwards curves can be computed by using these functions in Miller's algorithm. Computing the sum of two points or the double of a point and the coefficients of the corresponding functions is faster with our formulas than with all previously proposed formulas for pairings on Edwards curves. They are even competitive with all published formulas for pairing computation on Weierstrass curves. We also improve the formulas for Tate pairing computation on Weierstrass curves in Jacobian coordinates. Finally, we present several examples of pairing-friendly Edwards curves.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=nideQo-K9ME/.     

Efficient pairing computation on supersingular Abelian varieties

We present a general technique for the efficient computation of pairings on Jacobians of supersingular curves. This formulation, which we call the eta pairing, generalizes results of Duursma and Lee for computing the Tate pairing on supersingular elliptic curves in characteristic 3. We then show how our general technique leads to a new algorithm which is about twice as fast as the Duursma–Lee method. These ideas are applied to elliptic and hyperelliptic curves in characteristic 2 with very efficient results. In particular, the hyperelliptic case is faster than all previously known pairing algorithms.      

Poitou–Tate duality over extensions of global fields

In this paper, we are interested in the Poitou–Tate duality in Galois cohomology. We will formulate and prove a theorem for a nice class of modules (with a continuous Galois action) over a pro-p ring. The theorem will comprise of the Tate local duality, Poitou–Tate duality and the Poitou–Tate?s exact sequence.     

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou–Siebert. As an application, we determine a formula for such counts on P² and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in P² are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov–Witten theory. As a consequence, a new proof of Pandharipande’s formula for counts of elliptic curves in P² with fixed j-invariant is obtained.     

General stability of functional equations of linear type

Making use of a dynamical systems notion called shadowing, we prove a stability result for linear functional equations in metric groups. As a corollary we obtain stability of the quadratic functional equation in the case when the target space is a metric group satisfying some local 2-divisibility condition.