𝔽p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Daniele Bartoli; Maria Montanucci; Fernando Torres

doi:10.1515/advgeom-2021-0013

Published by De Gruyter June 28, 2021

𝔽_p²-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Daniele Bartoli , Maria Montanucci and Fernando Torres

From the journal Advances in Geometry

https://doi.org/10.1515/advgeom-2021-0013

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Abstract

Let 𝔽 be the finite field of order q². It is sometimes attributed to Serre that any curve 𝔽-covered by the Hermitian curveHq+1:yq+1=xq+x is also 𝔽-maximal. For prime numbers q we show that every 𝔽-maximal curve x of genus g ≥ 2 with | Aut(𝒳) | > 84(g − 1) is Galois-covered by Hq+1. The hypothesis on | Aut(𝒳) | is sharp, since there exists an 𝔽-maximal curve x for q = 71 of genus g = 7 with | Aut(𝒳) | = 84(7 − 1) which is not Galois-covered by the Hermitian curve H72.

Keywords: Finite field; maximal curve; Hermitian curve; Galois-covering

MSC 2010: Primary 11G; Secondary 14G

Funding statement: The third author would like to thank Università degli Studi di Perugia, for financial support received during his academic visit in January–February 2017; he also was partially suported by CNPq-Brazil (grant 308326/2014-8). This research was partially supported by the Ministry for Education, University and Research of Italy (MIUR), Project PRIN 2012 Geometrie di Galois e strutture di incidenza-Prot.N. 2012XZE22K_005, and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA-INdAM).

Communicated by: G. Korchmáros

Acknowledgements

The authors would like to thank Massimo Giulietti for numerous discussions on the topic which led to significant improvements.

Unfortunately, Fernando Torres passed away in May 2020 due to covid-19. He was an inspiring colleague for anyone who knew him. We dedicate this article to the memory of Fernando Torres.

References

[1] N. Arakelian, S. Tafazolian, F. Torres, On the spectrum for the genera of maximal curves over small fields. Adv. Math. Commun. 12 (2018), 143–149. MR3808220 Zbl 1414.9494910.3934/amc.2018009Search in Google Scholar

[2] D. Bartoli, M. Montanucci, F. Torres, Fp2− maximal curves with many automorphisms are Galois-covered by the Hermitian curve. Preprint 2018, arXiv 1708.03933v1 [math.AG]Search in Google Scholar

[3] D. Bartoli, M. Montanucci, G. Zini, AG codes and AG quantum codes from the GGS curve. Des. Codes Cryptogr. 86 (2018), 2315–2344. MR3845314 Zbl 1408.9499310.1007/s10623-017-0450-5Search in Google Scholar

[4] D. Bartoli, M. Montanucci, G. Zini, Multi point AG codes on the GK maximal curve. Des. Codes Cryptogr. 86 (2018), 161–177. MR3742839 Zbl 1400.9419410.1007/s10623-017-0333-9Search in Google Scholar

[5] B. W. Brock, Superspecial curves of genera two and three. PhD thesis, Princeton University 1993. MR2689446Search in Google Scholar

[6] A. Cossidente, G. Korchmáros, F. Torres, Curves of large genus covered by the Hermitian curve. Comm. Algebra 28 (2000), 4707–4728. MR1779867 Zbl 0974.1103110.1080/00927870008827115Search in Google Scholar

[7] W. L. Edge, A canonical curve of genus 7. Proc. London Math. Soc. (3) 17 (1967), 207–225. MR208460 Zbl 0196.5360210.1112/plms/s3-17.2.207Search in Google Scholar

[8] N. D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over ℚ. Invent. Math. 89 (1987), 561–567. MR903384 Zbl 0631.1402410.1007/BF01388985Search in Google Scholar

[9] N. D. Elkies, Shimura curve computations. In: Algorithmic number theory Portland, OR, 1998), volume 1423 of Lecture Notes in Comput. Sci., 1–47, Springer 1998. MR1726059 Zbl 1010.1103010.1007/BFb0054850Search in Google Scholar

[10] S. Fanali, M. Giulietti, On maximal curves with Frobenius dimension 3. Des. Codes Cryptogr. 53 (2009), 165–174. MR2545690 Zbl 1185.1104210.1007/s10623-009-9302-2Search in Google Scholar

[11] S. Fanali, M. Giulietti, I. Platoni, On maximal curves over finite fields of small order. Adv. Math. Commun. 6 (2012), 107–120. MR2885851 Zbl 1258.1107210.3934/amc.2012.6.107Search in Google Scholar

[12] R. Fricke, Ueber eine einfache Gruppe von 504 Operationen. Math. Ann. 52 (1899), 321–339. MR1511059 JFM 30.0143.0410.1007/BF01476163Search in Google Scholar

[13] R. Fuhrmann, A. Garcia, F. Torres, On maximal curves. J. Number Theory 67 (1997), 29–51. MR1485426 Zbl 0914.1103610.1006/jnth.1997.2148Search in Google Scholar

[14] R. Fuhrmann, F. Torres, The genus of curves over finite fields with many rational points. Manuscripta Math. 89 (1996), 103–106. MR1368539 Zbl 0857.1103210.1007/BF02567508Search in Google Scholar

[15] R. Fuhrmann, F. Torres, On Weierstrass points and optimal curves. Rend. Circ. Mat. Palermo (2) Suppl. no. 51 (1998), 25–46. MR1631013 Zbl 1049.11062Search in Google Scholar

[16] A. Garcia, Curves over finite fields attaining the Hasse–Weil upper bound. In: European Congress of Mathematics, Vol. II Barcelona, 2000), 199–205, Birkhäuser, Basel 2001. MR1905360 Zbl 1047.1106010.1007/978-3-0348-8266-8_15Search in Google Scholar

[17] A. Garcia, On curves with many rational points over finite fields. In: Finite fields with applications to coding theory, cryptography and related areas Oaxaca, 2001), 152–163, Springer 2002. MR1995333 Zbl 1045.1104510.1007/978-3-642-59435-9_11Search in Google Scholar

[18] A. Garcia, C. Güneri, H. Stichtenoth, A generalization of the Giulietti–Korchmáros maximal curve. Adv. Geom. 10 (2010), 427–434. MR2660419 Zbl 1196.1402310.1515/advgeom.2010.020Search in Google Scholar

[19] A. Garcia, H. Stichtenoth, Algebraic function fields over finite fields with many rational places. IEEE Trans. Inform. Theory 41 (1995), 1548–1563. MR1391016 Zbl 0863.1104010.1109/18.476212Search in Google Scholar

[20] A. Garcia, H. Stichtenoth, C.-P. Xing, On subfields of the Hermitian function field. Compositio Math. 120 (2000), 137–170. MR1739176 Zbl 0990.1104010.1023/A:1001736016924Search in Google Scholar

[21] A. Garcia, S. Tafazolian, Certain maximal curves and Cartier operators. Acta Arith. 135 (2008), 199–218. MR2457195 Zbl 1166.1101910.4064/aa135-3-1Search in Google Scholar

[22] M. Giulietti, J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Curves covered by the Hermitian curve. Finite Fields Appl. 12 (2006), 539–564. MR2257083 Zbl 1218.1106410.1016/j.ffa.2004.10.003Search in Google Scholar

[23] M. Giulietti, G. Korchmáros, A new family of maximal curves over a finite field. Math. Ann. 343 (2009), 229–245. MR2448446 Zbl 1160.1401610.1007/s00208-008-0270-zSearch in Google Scholar

[24] M. Giulietti, L. Quoos, G. Zini, Maximal curves from subcovers of the GK-curve. J. Pure Appl. Algebra 220 (2016), 3372–3383. MR3497966 Zbl 1417.1112010.1016/j.jpaa.2016.04.004Search in Google Scholar

[25] B. Gunby, A. Smith, A. Yuan, Irreducible canonical representations in positive characteristic. Res. Number Theory 1 (2015), Paper No. 3, 25. MR3500987 Zbl 1379.1401810.1007/s40993-015-0004-8Search in Google Scholar

[26] J. P. Hansen, Codes on the Klein quartic, ideals, and decoding. IEEE Trans. Inform. Theory 33 (1987), 923–925. MR923249 Zbl 0638.9401610.1109/TIT.1987.1057365Search in Google Scholar

[27] C. Heegard, J. Little, K. Saints, Systematic encoding via Gröbner bases for a class of algebraic-geometric Goppa codes. IEEE Trans. Inform. Theory 41 (1995), 1752–1761. MR1391033 Zbl 0857.9401510.1109/18.476247Search in Google Scholar

[28] R. A. Hidalgo, Edmonds maps on the Fricke–Macbeath curve. Ars Math. Contemp. 8 (2015), 275–289. MR3322704 Zbl 1338.3003310.26493/1855-3974.496.61aSearch in Google Scholar

[29] J. W. P. Hirschfeld, G. Korchmáros, F. Torres, Algebraic curves over a finite field. Princeton Univ. Press 2008. MR2386879 Zbl 1200.1104210.1515/9781400847419Search in Google Scholar

[30] Y. Ihara, Some remarks on the number of rational points of algebraic curves over finite fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 721–724 (1982). MR656048 Zbl 0509.14019Search in Google Scholar

[31] D. Joyner, An error-correcting codes package. SIGSAM Commun. Comput. Algebra 39(2) (2005), 65–68.10.1145/1101884.1101890Search in Google Scholar

[32] S. L. Kleiman, Algebraic cycles and the Weil conjectures. In: Dix exposés sur la cohomologie des schémas, volume 3 of Adv. Stud. Pure Math., 359–386, North-Holland 1968. MR292838 Zbl 0198.25902Search in Google Scholar

[33] F. Klein, Ueber die Transformation siebenter Ordnung der elliptischen Functionen. Math. Ann. 14 (1878), 428–471. MR1509988 JFM 11.0297.0110.1007/BF01677143Search in Google Scholar

[34] G. Korchmáros, F. Torres, On the genus of a maximal curve. Math. Ann. 323 (2002), 589–608. MR1923698 Zbl 1018.1102910.1007/s002080200316Search in Google Scholar

[35] G. Lachaud, Sommes d’Eisenstein et nombre de points de certaines courbes algébriques sur les corps finis. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), 729–732. MR920053 Zbl 0639.14013Search in Google Scholar

[36] L. Ma, C. Xing, On subfields of the Hermitian function field involving the involution automorphism. J. Number Theory 198 (2019), 293–317. MR3912940 Zbl 1426.1112910.1016/j.jnt.2018.10.014Search in Google Scholar

[37] A. M. Macbeath, On a curve of genus 7. Proc. London Math. Soc. (3) 15 (1965), 527–542. MR177342 Zbl 0146.4270510.1112/plms/s3-15.1.527Search in Google Scholar

[38] G. L. Matthews, Codes from the Suzuki function field. IEEE Trans. Inform. Theory 50 (2004), 3298–3302. MR2103499 Zbl 1316.9411910.1109/TIT.2004.838102Search in Google Scholar

[39] G. L. Matthews, Weierstrass semigroups and codes from a quotient of the Hermitian curve. Des. Codes Cryptogr. 37 (2005), 473–492. MR2177647 Zbl 1142.9439310.1007/s10623-004-4038-5Search in Google Scholar

[40] M. Montanucci, G. Zini, Some Ree and Suzuki curves are not Galois covered by the Hermitian curve. Finite Fields Appl. 48 (2017), 175–195. MR3705742 Zbl 1423.1111510.1016/j.ffa.2017.07.007Search in Google Scholar

[41] M. Montanucci, G. Zini, On the spectrum of genera of quotients of the Hermitian curve. Comm. Algebra 46 (2018), 4739–4776. MR3864261 Zbl 0695929210.1080/00927872.2018.1455100Search in Google Scholar

[42] H.-G. Rück, H. Stichtenoth, A characterization of Hermitian function fields over finite fields. J. Reine Angew. Math. 457 (1994), 185–188. MR1305281 Zbl 0802.11053Search in Google Scholar

[43] H. Stichtenoth, A note on Hermitian codes over GF(q²). IEEE Trans. Inform. Theory 34 (1988), 1345–1348. MR987682 Zbl 0665.9401510.1109/18.21267Search in Google Scholar

[44] H. Stichtenoth, Algebraic function fields and codes. Springer 2009. MR2464941 Zbl 1155.1402210.1007/978-3-540-76878-4Search in Google Scholar

[45] S. Tafazolian, A. Teherán-Herrera, F. Torres, Further examples of maximal curves which cannot be covered by the Hermitian curve. J. Pure Appl. Algebra 220 (2016), 1122–1132. MR3414410 Zbl 1401.1111110.1016/j.jpaa.2015.08.010Search in Google Scholar

[46] S. Tafazolian, F. Torres, A note on certain maximal curves. Comm. Algebra 45 (2017), 764–773. MR3562537 Zbl 1416.1110010.1080/00927872.2016.1175460Search in Google Scholar

[47] H. J. Tiersma, Remarks on codes from Hermitian curves. IEEE Trans. Inform. Theory 33 (1987), 605–609. MR901689 Zbl 0627.9401910.1109/TIT.1987.1057327Search in Google Scholar

[48] J. Top, C. Verschoor, Counting points on the Fricke–Macbeath curve over finite fields. J. Théor. Nombres Bordeaux 30 (2018), 117–129. MR3809712 Zbl 1408.1109310.5802/jtnb.1019Search in Google Scholar

[49] G. van der Geer, Coding theory and algebraic curves over finite fields: a survey and questions. In: Applications of algebraic geometry to coding theory, physics and computation Eilat, 2001), volume 36 of NATO Sci. Ser. II Math. Phys. Chem., 139–159, Kluwer 2001. MR1866898 Zbl 1011.1104210.1007/978-94-010-1011-5_8Search in Google Scholar

[50] G. van der Geer, Curves over finite fields and codes. In: European Congress of Mathematics, Vol. II Barcelona, 2000), 225–238, Birkhäuser, Basel 2001. MR1905363 Zbl 1025.1102210.1007/978-3-0348-8266-8_18Search in Google Scholar

[51] G. van der Geer, Counting curves over finite fields. Finite Fields Appl. 32 (2015), 207–232. MR3293411 Zbl 1378.1107110.1016/j.ffa.2014.09.008Search in Google Scholar

[52] C. Xing, H. Chen, Improvements on parameters of one-point AG codes from Hermitian curves. IEEE Trans. Inform. Theory 48 (2002), 535–537. MR1891265 Zbl 1071.9453610.1109/18.979330Search in Google Scholar

[53] C. Xing, S. Ling, A class of linear codes with good parameters from algebraic curves. IEEE Trans. Inform. Theory 46 (2000), 1527–1532. MR1768563 Zbl 1003.9403810.1109/18.850687Search in Google Scholar

[54] C. P. Xing, H. Stichtenoth, The genus of maximal function fields over finite fields. Manuscripta Math. 86 (1995), 217–224. MR1317746 Zbl 0826.1105410.1007/BF02567990Search in Google Scholar

[55] K. Yang, P. V. Kumar, On the true minimum distance of Hermitian codes. In: Coding theory and algebraic geometry Luminy, 1991), volume 1518 of Lecture Notes in Math., 99–107, Springer 1992. MR1186418 Zbl 0763.9402310.1007/BFb0087995Search in Google Scholar

Received: 2019-01-23

Revised: 2019-11-25

Published Online: 2021-06-28

Published in Print: 2021-07-27

𝔽_p²-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Abstract

Acknowledgements

References

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