Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Pickett, Erik Jarl

doi:10.1142/s1793042110003654

Cited by 5 publications

(8 citation statements)

References 23 publications

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“…We are grateful to Eric Pickett for alerting us to the results on integral normal bases contained in his paper [Pic10]. The main idea of this paper was developed during our visit to Salt Lake City in February 2011; we would like to thank Wieslawa Niziol and Pierre Colmez for their hospitality.…”

Section: Acknowledgementsmentioning

confidence: 98%

Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

Loeffler

Zerbes

2014

Int. J. Number Theory

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Abstract. We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of Qp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2.We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localisation at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a conjecture on the existence of a "zeta element", whose image under the regulator map is a p-adic L-function. We show that this conjecture implies the known properties of the 2-variable p-adic L-functions constructed by Perrin-Riou and Kim.

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Section: Acknowledgementsmentioning

confidence: 98%

Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

Loeffler

Zerbes

2014

Int. J. Number Theory

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“…Constructions of normal bases and self-dual bases have been extensively studied in the past two decades. A non exhaustive list of references is [10,11,12,13,14]. The latest results can be found for instance in [13] and [14], where explicit constructions of self-dual (integral) normal bases in abelian extensions of finite and local fields were given.…”

Section: Introductionmentioning

confidence: 99%

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Zhang¹,

Feng²,

Liao³

et al. 2014

Finite Fields and Their Applications

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Normal bases and self-dual normal bases over finite fields have been found to be very useful in many fast arithmetic computations. It is well-known that there exists a self-dual normal basis of F2n over F2 if and only if 4 ∤ n. In this paper, we prove there exists a normal element α of F2n over F2 corresponding to a prescribed vector a = (a0, a1, · · · , an−1) ∈ F n 2 such that ai = Tr 2 n |2 (α 1+2 i ) for 0 ≤ i ≤ n − 1, where n is a 2-power or odd, if and only if the given vector a is symmetric (ai = an−i for all i, 1 ≤ i ≤ n − 1), and one of the following is true. 1) n = 2 s ≥ 4, a0 = 1, a n/2 = 0,Furthermore we give an algorithm to obtain normal elements corresponding to prescribed vectors in the above two cases. For a general positive integer n with 4|n, some necessary conditions for a vector to be the corresponding vector of a normal element of F2n over F2 are given. And for all n with 4|n, we prove that there exists a normal element of F2n over F2 such that the Hamming weight of its corresponding vector is 3, which is the lowest possible Hamming weight.

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“…These authors point out that the cost of the exhaustive enumeration of the elements of F 2 n used to look for normal basis generators is a severe limitation to their method when the degree grows. On the other hand, their Table 4 shows that the minimal complexity for normal bases is very often reached by so-called self-dual bases (in all degrees not divisible by 4 up to 35 apart from 7,10,21). Restricting to self-dual normal bases enables one to push computations further; Geiselmann was indeed able to compute the lowest complexity for self-dual normal bases over F 2 up to degree 47 [10, loc.…”

Section: Introductionmentioning

confidence: 99%

“…Poli extended Wang's method to deal with the general characteristic 2 case in [22]. Recently, Pickett designed in [21] a construction that extends the former ones to the odd characteristic case, dealing separately with the semi-simple case and the ramified case.…”

Section: Introductionmentioning

confidence: 99%

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Construction of self-dual normal bases and their complexity

Arnault

Pickett

Vinatier

2012

Finite Fields and Their Applications

Self Cite

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Recent work of Pickett has given a construction of self-dual normal bases for extensions of finite fields, whenever they exist. In this article we present these results in an explicit and constructive manner and apply them, through computer search, to identify the lowest complexity of self-dual normal bases for extensions of low degree. Comparisons to similar searches amongst normal bases show that the lowest complexity is often achieved from a self-dual normal basis

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Construction of Self-Dual Integral Normal Bases in Abelian Extensions of Finite and Local Fields

Cited by 5 publications

References 23 publications

Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions

Finding normal bases over finite fields with prescribed trace self-orthogonal relations

Construction of self-dual normal bases and their complexity

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