Groups of Order 32 as Galois Groups

2000 Mathematics Subject Classification: 12F12.

Equivariant Birch-Swinnerton-Dyer Conjecture for the Base Change of Elliptic Curves: An Example

Let E be an elliptic curve defined over Q and let K/Q be a finite Galois extension with Galois group G. The equivariant Birch-Swinnerton-Dyer conjecture for h(1)(E x(Q) K)(1) viewed as amotive over Q with coefficients in Q[G] relates the twisted L-values associated with E with the arithmetic invariants of the same. In this paper I prescribe an approach to verify this conjecture for a given data. Using this approach, we verify the conjecture for an elliptic curve of conductor 11 and an S-3-extension of Q.

Computing Generators of Free Modules over Orders in Group Algebras

Let E be a number field and G be a finite group. Let A be any O_E-order of full rank in the group algebra E[G] and X be a (left) A-lattice. We give a necessary and sufficient condition for X to be free of given rank d over A. In the case that the Wedderburn decomposition E[G] \cong \oplus_xM_x is explicitly computable and each M_x is in fact a matrix ring over a field, this leads to an algorithm that either gives elements \alpha_1,...,\alpha_d \in X such that X = A\alpha_1 \oplus ... \oplusA\alpha_d or determines that no such elements exist. Let L/K be a finite Galois extension of number fields with Galois group G such that E is a subfield of K and put d = [K : E]. The algorithm can be applied to certain Galois modules that arise naturally in this situation. For example, one can take X to be O_L, the ring of algebraic integers of L, and A to be the associated order A(E[G];O_L) \subseteq E[G]. The application of the algorithm to this special situation is implemented in Magma under certain extra hypotheses when K = E = \IQ.

Algorithms for Tamagawa Number Conjectures

In dieser Arbeit werden Algorithmen zur Untersuchung der äquivarianten Tamagawazahlvermutung von Burns und Flach entwickelt. Zunächst werden Algorithmen angegeben mit denen die lokale Fundamentalklasse, die globale Fundamentalklasse und Tates kanonische Klasse berechnet werden können. Dies ermöglicht unter anderem Berechnungen in Brauergruppen von Zahlkörpererweiterungen. Anschließend werden diese Algorithmen auf die Tamagawazahlvermutung angewendet. Die Epsilonkonstantenvermutung kann dadurch für alle Galoiserweiterungen L|K bewiesen werden, bei denen L in einer Galoiserweiterung E|Q vom Grad kleiner gleich 15 eingebettet werden kann. Für die Tamagawazahlvermutung an der Stelle 1 wird ein Algorithmus angegeben, der die Vermutung für ein gegebenes Fallbeispiel L|Q numerischen verifizieren kann. Im Spezialfall, dass alle Charaktere rational oder abelsch sind, kann dieser Algorithmus die Vermutung für L|Q sogar beweisen.

Construction and decoding of BCH codes over finite commutative rings

BCH codes over arbitrary finite commutative rings with identity are derived in terms of their locator vector. The derivation is based on the factorization of xs -1 over the unit ring of an appropriate extension of the finite ring. We present an efficient decoding procedure, based on the modified Berlekamp-Massey algorithm, for these codes. The code construction and the decoding procedures are very similar to the BCH codes over finite integer rings. © 1999 Elsevier B.V. All rights reserved.

Forma traço sobre algumas extensões galoisianas de corpos p-Ádicos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Henselian Discrete Valued Fields Admitting One-Dimensional Local Class Field Theory

2000 Mathematics Subject Classification: 11S31 12E15 12F10 12J20.

On realizability of p-groups as Galois groups

2000 Mathematics Subject Classification: 12F12, 15A66.

Type extension and efficient AST manipulation

Oberon-2 is an object-oriented language with a class structure based on type extension. The runtime structure of Oberon-2 is described and the low-level mechanism for dynamic type checking explained. It is shown that the superior type-safety of the language, when used for programming styles based on heterogeneous, pointer-linked data structures, has an entirely negligible cost in runtime performance.

Avoiding full extension field arithmetic in pairing computations

The most costly operations encountered in pairing computations are those that take place in the full extension field Fpk . At high levels of security, the complexity of operations in Fpk dominates the complexity of the operations that occur in the lower degree subfields. Consequently, full extension field operations have the greatest effect on the runtime of Miller’s algorithm. Many recent optimizations in the literature have focussed on improving the overall operation count by presenting new explicit formulas that reduce the number of subfield operations encountered throughout an iteration of Miller’s algorithm. Unfortunately, almost all of these improvements tend to suffer for larger embedding degrees where the expensive extension field operations far outweigh the operations in the smaller subfields. In this paper, we propose a new way of carrying out Miller’s algorithm that involves new explicit formulas which reduce the number of full extension field operations that occur in an iteration of the Miller loop, resulting in significant speed ups in most practical situations of between 5 and 30 percent.