This essay starts by first explaining, for elliptic curves defined over Q, the statement of the conjecture of Birch and Swinnerton-Dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the ess...

Later on in [2], Grekos refined the previous results, in some cases, by introducing the infimum of the radii of curvature r(Γ ) of the curve. He succeeded in showing an upper bound of the shape N(Γ ) l(Γ )r(Γ )−1/3 and conversely constructed a family of curves Γ0 with N(Γ0) l(Γ0)r(Γ0)−1/3. Naturally, Grekos’ results suppose that the curve has at least C2-regularity. In fact, this is the maximal...

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...

Abstract We show that the conjecture of [27] for special value at $s=1$ zeta function an arithmetic surface is equivalent to Birch–Swinnerton–Dyer Jacobian generic fibre.

R. W. Bagley, E. G. Begle, J. W. Brace, A. T. Brauer, E. A. Cameron, C. ECapel, Haskell Cohen, H. S. Collins, F. E. Cothran, A. C. Downing, Jr., B. M. Drucker, L. A. Dye, S. E. Dyer, Jr., J. C. Eaves, D. O. Ellis, M. H. M. Esser, W. M. Faucett, F. A* Ficken, L. R. Ford, L. R. Ford, Jr., M. K. Fort, Jr., Tomlinson Fort, H. K. Fulmer, Seymour Ginsburg, Wallace Givens, R. A. Good, W. L. Gordon, E....

at the SPTL conference, Glasgow University 2002. I would like to thank Sara Fovargue for her comments on that draft and the anonymous reviewers of Medical Law Review. 1 See further C. Dyer, ‘Surrogate Mother Refuses to Give Up Baby’ (1997) 314 B.M.J. 250 and D. Kennedy, ‘Minister Hints at Change in the Law’ (1997) The Times, 16 May. 2 M. Brazier, A. Campbell and S. Golombok, Surrogacy: Review f...

Dyer-Lashof operations were first introduced by Araki and Kudo in [1] in order to calculate ü*(QS+; Z2). These operations were later used by Dyer and Lashof to determine H*(QY;ZP) as a functor of H*(Y;ZP) [5], where QY = | J n H n E n y . This has had many important applications. Hodgkin and Snaith independently defined a single secondary operation in if-homology (for p odd and p = 2 respective...

In [8], Swinnerton-Dyer considered the proportion of twists of an elliptic curve with full 2-torsion that have 2-Selmer group of a particular dimension. Swinnerton-Dyer obtained asymptotic results on the number of such twists using an unusual notion of asymptotic density. We build on this work to obtain similar results on the density of twists with particular rank of 2-Selmer group using the na...

The aim of this study is to evaluate the reliability of a crowd simulation model developed by the authors by reproducing Dyer et al.'s experiments (published in Philosophical Transactions in 2009) on human leadership and consensus decision making in a computer-based environment. The theoretical crowd model of the simulation environment is presented, and its results are compared and analysed aga...

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