Let E be an optimal elliptic curve of conductor N , such that the L-function LE(s) of E vanishes to order one at s = 1. Let K be a quadratic imaginary field in which all the primes dividing N split. The Gross-Zagier theorem gives a formula that expresses the Birch and Swinnerton-Dyer conjectural order the Shafarevich-Tate group of E over K as a rational number. We extract an integer factor from...

Recently, the authors proposed a method for computing the Tate pairing using a distortion map for y2 = x5 − αx (α = ±2) over finite fields of characteristic five. In this paper, we show the Ate pairing, an invariant of the Tate pairing, can be applied to this curve. This leads to about 50% computational cost-saving over the Tate pairing.

The problem of estimating the number of imaginary quadratic fields whose ideal class group has an element of order ` ≥ 2 is classical in number theory. Analogous questions for quadratic twists of elliptic curves have been the focus of recent interest. Whereas works of Stewart and Top [St-T], and of Gouvêa and Mazur [G-M] address the nontriviality of MordellWeil groups, less is known about the n...

The theory of Pell’s equation has a long history, as can be seen from the huge amount of references collected in Dickson [Dic1920], from the two books on its history by Konen [Kon1901] and Whitford [Whi1912], or from the books by Weber [Web1939], Walfisz [Wal1952], Faisant [Fai1991], and Barbeau [Bar2003]. For the better part of the last few centuries, the continued fractions method was the und...

CLASS FIELD THEORY: CLASS FORMATIONS Tate’s theorem (see (4) above) shows that, in order to have a class field theory over a field k, all one needs is, for each system of fields ksep L K k; ŒLWk <1; L=K Galois, a G.L=K/-module CL and a “fundamental class” uL=K 2 H .G.L=K/;CL/ satisfying Tate’s hypotheses; the pairs .CL;uL=K/ should also satisfy certain natural conditions when K and L vary. The...

For the Tate pairing implementation over hyperelliptic curves, there is a development by DuursmaLee and Barreto et al., and those computations are focused on degenerate divisors. As divisors are not degenerate form in general, it is necessary to find algorithms on general divisors for the Tate pairing computation. In this paper, we present two efficient methods for computing the Tate pairing ov...

We describe the Hodge theory of brilliant families K3 surfaces. Their characteristic feature is a close link between structures any two fibres over points in Noether–Lefschetz locus. Twistor deformations, analytic Tate–Šafarevič group, and certain one-dimensional Shimura special cycles are covered by theory. In this setting, Brauer group viewed as locus family or specialization loci approaching...

Let E be an abelian variety deened over a number eld K. Let p be a prime number. Let X(K;E) p 1 be the p-Tate-Shafarevich group of E and S class E p 1 (K) the p 1-Selmer group of E. Thirty years ago, Tate proved a local duality theorem for E and used it to establish a global duality for E, later called Cassels-Tate pairing 3, 14]. It states that there is a pairing between X(K;E) p 1 and the p-T...

We discuss abelian equivariant Iwasawa theory for elliptic curves over $${\mathbb {Q}}$$ at good supersingular primes and non-anomalous ordinary primes. Using Kobayashi’s method, we construct Coleman maps, which send the Beilinson–Kato element to p-adic L-functions. Then propose main conjectures and, under certain assumptions, prove one divisibility via Euler system machinery. As an application...

Purpose: Somatostatin receptors (SSTr) are expressed on many neuroendocrine tumors, and several radiotracers have been developed for imaging these types of tumors. For this reason, peptide analogues of somatostatin have been well characterized. Copper-64 (t1/2 12.7 hours), a positron emitter suitable for positron emission tomography (PET) imaging, was shown recently to have improved in vivo cle...