Maximal curves and Tate-Shafarevich results for quartic and sextic twists. (English) Zbl 1519.14024
Summary: We study elliptic surfaces corresponding to an equation of the specific type \(y^2 = x^3 + f(t) x\), defined over the finite field \(\mathbb{F}_q\) for a prime power \(q \equiv 3 \bmod 4\). It is shown that if \(s^4 = f(t)\) defines a curve that is maximal over \(\mathbb{F}_{q^2}\) then the rank of the group of sections defined over \(\mathbb{F}_q\) on the elliptic surface is determined in terms of elementary properties of the rational function \(f(t)\). Similar results are shown for elliptic surfaces given by \(y^2 = x^3 + g(t)\) using prime powers \(q \equiv 5 \operatorname{mod} 6\) and curves \(s^6 = g(t)\). Finally, for each of the forms used here, existence of curves with the property that they are maximal over \(\mathbb{F}_{q^2}\) is discussed, as well as various examples.
MSC:
| 14G15 | Finite ground fields in algebraic geometry |
| 11G20 | Curves over finite and local fields |
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
| 14H25 | Arithmetic ground fields for curves |
Keywords:
finite field; maximal curve; function field; elliptic curve; elliptic surface; Mordell-Weil rankSoftware:
manYPointsReferences:
| [1] | Anbar, Nurdagül; Bassa, Alp; Beelen, Peter, A complete characterization of Galois subfields of the generalized Giulietti-Korchmáros function field, Finite Fields Appl., 48, 318-330 (2017) · Zbl 1398.11096 |
| [2] | Arakelian, Nazar; Tafazolian, Saeed; Torres, Fernando, On the spectrum for the genera of maximal curves over small fields, Adv. Math. Commun., 12, 143-149 (2018) · Zbl 1414.94949 |
| [3] | Beelen, Peter; Landi, Leonardo; Montanucci, Maria, Classification of all Galois subcovers of the Skabelund maximal curves, J. Number Theory, 242, 46-72 (2023) · Zbl 1508.14030 |
| [4] | Bouw, Irene I.; Diem, Claus; Scholten, Jasper, Ordinary elliptic curves of high rank over \(\overline{\mathbb{F}_p}(x)\) with constant j-invariant, Manuscr. Math., 114, 487-501 (2004) · Zbl 1128.11029 |
| [5] | Diem, Claus; Scholten, Jasper, Ordinary elliptic curves of high rank over \(\overline{\mathbb{F}_p}(x)\) with constant j-invariant. II, J. Number Theory, 124, 31-41 (2007) · Zbl 1190.11033 |
| [6] | Fanali, Stefania; Giulietti, Massimo; Platoni, Irene, On maximal curves over finite fields of small order, Adv. Math. Commun., 6, 107-120 (2012) · Zbl 1258.11072 |
| [7] | Fuhrmann, Rainer; Garcia, Arnaldo; Torres, Fernando, On maximal curves, J. Number Theory, 67, 29-51 (1997) · Zbl 0914.11036 |
| [8] | Garcia, Arnaldo; Stichtenoth, Henning, On Chebyshev polynomials and maximal curves, Acta Arith., 90, 301-311 (1999) · Zbl 0933.11032 |
| [9] | Garcia, Arnaldo; Stichtenoth, Henning; Xing, Chao-Ping, On subfields of the Hermitian function field, Compos. Math., 120, 137-170 (2000) · Zbl 0990.11040 |
| [10] | van der Geer, Gerard; Howe, Everett W.; Lauter, Kristin E.; Ritzenthaler, Christophe, Tables of curves with many points (2009), Retrieved November 28th, 2022 |
| [11] | Hirschfeld, J. W.P.; Korchmáros, G.; Torres, F., Algebraic Curves over a Finite Field, Princeton Series in Applied Mathematics (2008), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 1200.11042 |
| [12] | Kazemifard, Ahmad; Tafazolian, Saeed; Torres, Fernando, On maximal curves related to Chebyshev polynomials, Finite Fields Appl., 52, 200-213 (2018) · Zbl 1443.11118 |
| [13] | Lachaud, Gilles, Sommes d’Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis, C. R. Acad. Sci. Paris, Sér. I Math., 305, 729-732 (1987) · Zbl 0639.14013 |
| [14] | Mendoza, Erik A. R.; Quoos, Luciane, Explicit equations for maximal curves as subcovers of the BM curve, Finite Fields Appl., 77, Article 101945 pp. (2022) · Zbl 1475.11126 |
| [15] | Schütt, Matthias; Shioda, Tetsuji, Mordell-Weil Lattices, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 70 (2019), Springer: Springer Singapore · Zbl 1433.14002 |
| [16] | Serre, Jean-Pierre, Rational Points on Curves over Finite Fields, Documents Mathématiques (Paris), vol. 18 (2020), Société Mathématique de France: Société Mathématique de France Paris, With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler, Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoof · Zbl 1475.11002 |
| [17] | Shioda, Tetsuji, An explicit algorithm for computing the Picard number of certain algebraic surfaces, Am. J. Math., 108, 415-432 (1986) · Zbl 0602.14033 |
| [18] | Tafazolian, Saeed; Top, Jaap, On certain maximal hyperelliptic curves related to Chebyshev polynomials, J. Number Theory, 203, 276-293 (2019) · Zbl 1444.11134 |
| [19] | Tafazolian, Saeed; Torres, Fernando, On the curve \(y^n = x^m + x\) over finite fields, J. Number Theory, 145, 51-66 (2014) · Zbl 1377.11075 |
| [20] | Tate, John T., Endomorphisms of abelian varieties over finite fields, Invent. Math., 2, 134-144 (1966) · Zbl 0147.20303 |
| [21] | Tate, John T.; Shafarevich, Igor R., The rank of elliptic curves, Dokl. Akad. Nauk SSSR, 175, 770-773 (1967) · Zbl 0168.42201 |
| [22] | Top, Jaap, Hecke L-series related with algebraic cycles or with Siegel modular forms (1989), University of Utrecht, PhD thesis |
| [23] | Top, Jaap, Descent by 3-isogeny and 3-rank of quadratic fields, (Advances in Number Theory. Advances in Number Theory, Kingston, ON, 1991. Advances in Number Theory. Advances in Number Theory, Kingston, ON, 1991, Oxford Sci. Publ. (1993), Oxford Univ. Press: Oxford Univ. Press New York), 303-317 · Zbl 0804.11040 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
