On the zeta functions of an optimal tower of function fields over 𝔽₄

AbstractWe give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

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Bases for Riemann–Roch Spaces of One-Point Divisors on an Optimal Tower of Function Fields

IEEE Transactions on Information Theory ◽

10.1109/tit.2011.2179519 ◽

2012 ◽

Vol 58 (5) ◽

pp. 2589-2598 ◽

Cited By ~ 2

Author(s):

Francesco Noseda ◽

Gilvan Oliveira ◽

Luciane Quoos

Keyword(s):

Function Fields ◽

Tower Of Function Fields

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ON INVARIANTS OF TOWERS OF FUNCTION FIELDS OVER FINITE FIELDS

Journal of Algebra and Its Applications ◽

10.1142/s0219498812501903 ◽

2013 ◽

Vol 12 (04) ◽

pp. 1250190 ◽

Cited By ~ 1

Author(s):

FLORIAN HESS ◽

HENNING STICHTENOTH ◽

SEHER TUTDERE

Keyword(s):

Finite Field ◽

Finite Fields ◽

Function Fields ◽

Finite Extension ◽

Towers Of Function Fields ◽

Asymptotic Number ◽

Tower Of Function Fields

In this paper, we consider a tower of function fields [Formula: see text] over a finite field 𝔽q and a finite extension E/F0 such that the sequence [Formula: see text] is a tower over the field 𝔽q. Then we study invariants of [Formula: see text], that is, the asymptotic number of the places of degree r in [Formula: see text], for any r ≥ 1, if those of [Formula: see text] are known. We first give a method for constructing towers of function fields over any finite field 𝔽q with finitely many prescribed invariants being positive. For q a square, we prove that with the same method one can also construct towers with at least one positive invariant and certain prescribed invariants being zero. Our method is based on explicit extensions. Moreover, we show the existence of towers over a finite field 𝔽q attaining the Drinfeld–Vladut bound of order r, for any r ≥ 1 with qr a square (see [1, Problem-2]). Finally, we give some examples of non-optimal recursive towers with all but one invariants equal to zero.

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