In [15] and [16] all possible torsion groups of elliptic curves E with integral j-invariant over quadratic and pure cubic number fields K are determined. Moreover, with the exception of the torsion groups of isomorphism types ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ×ℤ/2ℤ, all elliptic curves E and all basic quadratic and pure cubic fields K such that E over K has one of these torsion groups were computed. The present paper is aimed at solving the corresponding problem for general cubic number fields K. In the general cubic case, the above groups ℤ/2ℤ, ℤ/3ℤ and ℤ/2ℤ×ℤ/2ℤ and, in addition, the groups ℤ/4ℤ, ℤ/5ℤ occur as torsion groups of infinitely many curves E with integral j-invariant over infinitely many cubic fields K. For all the other possible torsion groups, the (finitely any) elliptic curves with integral j over the (finitely many) cubic fields K are calculated here. Of course, the results obtained in [6] for pure cubic fields and in [24] for cyclic cubic fields are regained by our algorithms. However, compared with [15] and [6], a solution of the torsion group problem in the much more involved general cubic case requires some essentially new methods. In fact we shall use Gröbner basis techniques and elimination theory to settle the general case.
The ENC project attempts to realize an electron-nucleon collider at the upcoming Facility for Antiproton and Ion Research FAIR at GSI Darmstadt by utilizing the antiproton high-energy storage ring HESR for polarized proton and deuteron beams. The addition of a 3.3 GeV storage ring for polarized electrons will enable electron-nucleon collisions up to a center-of-mass energy of √ s ≈ 14 GeV. In such a configuration peak luminosities in the range of L = 10 32 to 10 33 cm −2 s −1 are feasible. Beam-beam effects in a space-charge dominated regimes in conjunction with high-energy electron cooling represents one of the main challenges for this project. In this paper beam-and spin dynamics simulations are presented, together with the required modifications and extensions for a collision mode of the HESR storage ring and the conceptual design of this new collider complex.
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