(−1)(1!2! · · · (n− 1)!) σ(nu) σ(u)n = ∣∣∣∣∣∣∣ ℘ ℘ · · · ℘ ℘ ℘ · · · ℘ .. .. . . . .. ℘ ℘ · · · ℘ ∣∣∣∣∣∣∣ (u). (0.2) Although this formula can be obtained by a limiting process from (0.1), it was found before [FS] by the paper of Kiepert [K]. If we set y(u) = 1 2℘ (u) and x(u) = ℘(u), then we have an equation y(u) = x(u)+ · · · , that is a defining equation of the elliptic curve to which the fu...

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

We introduce an elliptic analogue of the Apostol sums, which we call elliptic Apostol sums. These sums are defined by means of certain elliptic functions with a complex parameter τ having positive imaginary part. When τ → i∞, these elliptic Apostol sums represent the well-known Apostol generalized Dedekind sums. Also these elliptic Apostol sums are modular forms in the variable τ . We obtain a ...

We derive a general expression for an interface parameter which makes possible the design of a neutral elliptic inhomogeneity when the stress field in the surrounding matrix is a polynomial function of nth order and the composite is subjected to antiplane shear deformations. © 2005 Elsevier Ltd. All rights reserved.

Denote C (d) ∆ the relative symmetric product of this family over ∆ , parameterizing effective divisors of degree d on fibres Ct for t 6= 0. We will construct a compactification H̃d of C (d) ∆ over ∆, such that H̃d has smooth total space and the fibre over t = 0 has simple normal crossing support. By studying the fibre over t = 0 of H̃d, we understand how the symmetric products of smooth curves de...

To help motivate the Weil pairing, we discuss it in the context of elliptic curves over the field of complex numbers.

This is an exposition of some of the main features of the theory of elliptic curves and modular forms.

We show that the Hankel determinants of a generalized Catalan sequence satisfy the equations of the elliptic sequence. As a consequence, the coordinates of the multiples of an arbitrary point on the elliptic curve are expressed by the Hankel determinants. PACS numbers: 02.30.Ik, 02.30.Gp, 02.30.Lt

We prove Atiyah's classi cation results about indecomposable vector bundles on an elliptic curve by applying the Fourier-Mukai transform. We extend our considerations to semistable bundles and construct the universal stable sheaves. MSC 2000: 14H60 Vector bundles on curves and their moduli, 14H52 Elliptic curves.

For r = 6, 7, . . . , 11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r = 6) to over 100 (for r = 10 and r = 11). We describe our search methods, and tabulate, for each r = 5, 6, . . . , 11, the five curves of lowest conductor, and (except for r = 11) also the five of lowest absolute disc...