Tabulation of cubic function fields via polynomial binary cubic forms

Rozenhart, Pieter; Jacobson, Michael J.; Scheidler, Renate

doi:10.1090/s0025-5718-2012-02591-9

Cited by 5 publications

(16 citation statements)

References 28 publications

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“…Hessian contains a unique reduced representative, and there are only finitely many reduced binary cubic forms of any given discriminant with imaginary or unusual Hessian. The tabulation of F q (t)-isomorphism classes of cubic function fields as performed in [26,27,28] used the Davenport-Heilbronn bijection between F q (t)isomorphism classes of cubic function fields and a certain collection U of GL 2 (F q [t])isomorphism classes of binary cubic forms. This set U includes all classes of primitive, irreducible binary cubic forms with square-free discriminant, which is all that is required in our context.…”

Section: Every Equivalence Class Of Binary Cubic Forms With Imaginarymentioning

confidence: 99%

“…We now briefly describe our method for tabulating quadratic function fields of imaginary or unusual fundamental discriminant −3D with positive 3-rank up to a given bound B on deg(D). The basic algorithm builds on the algorithm for tabulating cubic function fields from [26,27,28], except instead of outputting minimal polynomials for all fields, we simply increment a counter for each square-free discriminant found. The counter and corresponding discriminant values are then output.…”

Section: The Algorithm and Its Complexitymentioning

confidence: 99%

“…The asymptotic complexity of the algorithm for generating fields with positive 3-rank is the same as for the tabulation algorithm for cubic function fields, namely O(B 4 q B ) field operations [27], with the O-constant cubic in q when B is odd and quartic in q when B is even. The main difference between the two algorithms is that instead of testing if an equivalence class of binary cubic forms belongs to the Davenport-Heilbronn set, we test if it has square-free discriminant.…”

Section: The Algorithm and Its Complexitymentioning

confidence: 99%

“…As before, let p denote the characteristic of F q . As in [27], denote by Then N (m, n, r) = r r − 1 mr n + O(r n ) as m → ∞.…”

Section: Complexity Of the Improved Algorithmmentioning

confidence: 99%

“…Our contributions are as follows. First, our basic algorithm (Algorithm 3.1) functions in much the same way as that in [27]: we loop over the coefficients of a reduced binary cubic form except instead of outputting a binary cubic form of discriminant D, we increment a counter for that discriminant value and store both D and its counter value in a hash table. The hash table is then output once the for loops are exited.…”

Section: Introductionmentioning

confidence: 99%

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Computing quadratic function fields with high 3-rank via cubic field tabulation

Rozenhart

Jacobson

Scheidler

2015

Rocky Mountain J. Math.

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This paper presents an algorithm for generating all imaginary and unusual discriminants up to a fixed degree bound that define a quadratic function field of positive 3-rank. Our method makes use of function field adaptations of a method due to Belabas for finding quadratic number fields of high 3-rank and of a refined function field version of a theorem due to Hasse. We provide numerical data for discriminant degree up to 11 over the finite fields F5, F7, F11 and F13. A special feature of our technique is that it produces quadratic function fields of minimal genus for any given 3-rank. Taking advantage of certain Fq(t)-automorphisms in conjunction with Horner's rule for evaluating polynomials significantly speeds up our algorithm in the imaginary case; this improvement is unique to function fields and does not apply to number field tabulation. These automorphisms also account for certain divisibility properties in the number of fields found with positive 3-rank. Our numerical data mostly agrees with the predicted heuristics of Friedman-Washington and partial results on the distribution of such values due to Ellenberg-Venkatesh-Westerland for quadratic function fields over the finite field Fq where q ≡ −1 (mod 3). The corresponding data for q ≡ 1 (mod 3) does not agree closely with the previously mentioned heuristics and results, but does agree more closely 1 arXiv:1003.1287v3 [math.NT] 3 May 2012 with some recent number field conjectures of Malle and some work in progress on proving such conjectures for function fields due to Garton.

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Section: Every Equivalence Class Of Binary Cubic Forms With Imaginarymentioning

confidence: 99%

Section: The Algorithm and Its Complexitymentioning

confidence: 99%

Section: The Algorithm and Its Complexitymentioning

confidence: 99%

“…As before, let p denote the characteristic of F q . As in [27], denote by Then N (m, n, r) = r r − 1 mr n + O(r n ) as m → ∞.…”

Section: Complexity Of the Improved Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Computing quadratic function fields with high 3-rank via cubic field tabulation

Rozenhart

Jacobson

Scheidler

2015

Rocky Mountain J. Math.

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Construction of all cubic function fields of a given square-free discriminant

Jacobson

Lee

Scheidler

et al. 2015

Int. J. Number Theory

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For any square-free polynomial D over a finite field of characteristic at least 5, we present an algorithm for generating all cubic function fields of discriminant D. We also provide a count of all these fields according to their splitting at infinity. When D′ = D/(-3) has even degree and a leading coefficient that is a square, i.e. D′ is the discriminant of a real quadratic function field, this method makes use of the infrastructures of this field. This infrastructure method was first proposed by Shanks for cubic number fields in an unpublished manuscript from the late 1980s. While the mathematical ingredients of our construction are largely classical, our algorithm has the major computational advantage of finding very small minimal polynomials for the fields in question.

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Cubic function fields with prescribed ramification

Karemaker

Marques

Sijsling

2021

Int. J. Number Theory

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This paper describes cubic function fields [Formula: see text] with prescribed ramification, where [Formula: see text] is a rational function field. We give general equations for such extensions, an explicit procedure to obtain a defining equation when the purely cubic closure [Formula: see text] of [Formula: see text] is of genus zero, and a description of the twists of [Formula: see text] up to isomorphism over [Formula: see text]. For cubic function fields of genus at most one, we also describe the twists and isomorphism classes obtained when one allows Möbius transformations on [Formula: see text]. The paper concludes by studying the more general case of covers of elliptic and hyperelliptic curves that are ramified above exactly one point.

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Tabulation of cubic function fields via polynomial binary cubic forms

Cited by 5 publications

References 28 publications

Computing quadratic function fields with high 3-rank via cubic field tabulation

Computing quadratic function fields with high 3-rank via cubic field tabulation

Construction of all cubic function fields of a given square-free discriminant

Cubic function fields with prescribed ramification

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