×

The scheme of monogenic generators. I: Representability. (English) Zbl 1524.11188

Summary: This is the first in a series of two papers that study monogenicity of number rings from a moduli-theoretic perspective. Given an extension of algebras \(B/A\), when is \(B\) generated by a single element \(\theta\in B\) over \(A\)? In this paper, we show there is a scheme \(\mathcal{M}_{B/A}\) parameterizing the choice of a generator \(\theta\in B\), a “moduli space” of generators. This scheme relates naturally to Hilbert schemes and configuration spaces. We give explicit equations and ample examples.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
14D20 Algebraic moduli problems, moduli of vector bundles
13E15 Commutative rings and modules of finite generation or presentation; number of generators
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alpöge, L., Bhargava, M., Shnidman, A.: A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so. (2020). arXiv: 2011.01186 [math.NT]
[2] Alpöge, L., Bhargava, M., Shnidman, A.: A positive proportion of quartic fields are not monogenic yet have no local obstruction to being so. (2021). arXiv: 2107.05514 [math.NT]
[3] Alling, NL, A proof of the corona conjecture for finite open Riemann surfaces, Bull. Am. Math. Soc., 70, 110-112 (1964) · Zbl 0124.04202 · doi:10.1090/S0002-9904-1964-11040-5
[4] Atiyah, M.F., Macdonald, I.G.: Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, pp. ix+128 (1969) · Zbl 0175.03601
[5] Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. In: J. Symbolic Comput. 24.3-4 (1997). Computational algebra and number theory (London, 1993), pp. 235-265. ISSN: 0747-7171. doi:10.1006/jsco.1996.0125 · Zbl 0898.68039
[6] Bérczes, A., Evertse, J.-H., Györy, K.: Multiply monogenic orders. In: Ann. Sc. Norm. Super. Pisa Cl. Sci. 5:12.2, pp. 467-497 (2013). ISSN: 0391-173X · Zbl 1319.11070
[7] Bhatt, B.: Algebraization and Tannaka duality. In: Cambridge Journal of Mathematics, vol. 4.4, pp. 403-461 (2016) · Zbl 1356.14006
[8] Bhargava, M.: On the number of monogenizations of a quartic order (2021). arXiv: 2111.04215 [math.NT]
[9] Bhargava, M., Hanke, J., Shankar, A.: The mean number of 2-torsion elements in the class groups of n-monogenized cubicfields (2020). arXiv: 2010.15744 [math.NT]
[10] Bosch, S., Lütkebohmert, W., Raynaud, M.: Néron Models. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Springer Berlin (2012). ISBN: 9783642514388. https://books.google.com/books?id=BNfnCAAAQBAJ
[11] Bhargava, M., Shankar, A., Wang, X.: Squarefree values of polynomial discriminants I. arXiv e-prints (2016). arXiv:1611.09806 [math.NT]
[12] Casnati, G., Ekedahl, T.: Covers of algebraic varieties. I. A general structure theorem, covers of degree 3; 4 and Enriques surfaces. In: J. Algebraic Geom., vol. 5.3, pp. 439-460 (1996). ISSN: 1056-3911 · Zbl 0866.14009
[13] Connett, JE, A generalization of the Borsuk-Ulam theorem, J. Lond. Math. Soc., 7, 64-66 (1973) · Zbl 0273.55013 · doi:10.1112/jlms/s2-7.1.64
[14] Dedekind, R.: Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen. In: Gött. Abhandlungen, pp. 1-23 (1878)
[15] Demailly, J.-P.: Algebraic criteria for kobayashi hyperbolic projective varieties and jet differentials, vol. 62, pp. 32-20 (1991). doi:10.1090/pspum/062.2/1492539
[16] Duvall, P.F., Husch, L. S.: Embedding coverings into bundles with applications. Memoirs Am. Math. Soc. 263 (1982) · Zbl 0525.57012
[17] Duchamp, T., Primitive elements in rings of holomorphic functions, J. für die reine und Angewandte Mathematik, 346, 199-220 (1984) · Zbl 0513.32004 · doi:10.1515/crll.1984.346.199
[18] Evertse, J.-H., Györy, K.: Discriminant equations in Diophantine number theory, vol. 32. New Mathematical Monographs. Cambridge University Press, Cambridge (2017), pp. xviii+457. ISBN: 978-1-107-09761-2. doi:10.1017/CBO9781316160763 · Zbl 1361.11002
[19] Evertse, J-H; Györy, K., On unit equations and decomposable form equations, J. Reine Angew. Math., 358, 6-19 (1985) · Zbl 0552.10010
[20] Evertse, J-H, A survey on monogenic orders, Publ. Math. Debrecen, 79, 3-4, 411-422 (2011) · Zbl 1249.11102 · doi:10.5486/PMD.2011.5150
[21] Fantechi, B., Fundamental Algebraic Geometry: Grothendieck’s FGA Explained (2005), Providence: AMS, Providence · Zbl 1085.14001
[22] Farrell, TF, Right-orderable deck transformation groups, Rocky Mt. J. Math., 6, 441-447 (1976) · Zbl 0332.57004 · doi:10.1216/RMJ-1976-6-3-441
[23] Gaál, I., Power integral bases in cubic relative extensions, Exp. Math., 10, 1, 133-139 (2001) · Zbl 1014.11080 · doi:10.1080/10586458.2001.10504436
[24] Gaál, I.: Diophantine equations and power integral bases. Theory and algorithms, 2nd edn [ MR1896601]. Birkhäuser/Springer, Cham (2019), pp. xxii+326. ISBN: 978-3-030-23864-3; 978-3-030-23865- 0. doi:10.1007/978-3-030-23865-0 · Zbl 1465.11090
[25] Gorin, EA; Lin, VJ, Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids, Math. USSR-Sbornik, 7, 569-596 (1969) · Zbl 0211.54905 · doi:10.1070/SM1969v007n04ABEH001104
[26] Gaál, I.; Pohst, M., Computing power integral bases in quartic relative extensions, J. Number Theory, 85, 2, 201-219 (2000) · Zbl 0993.11055 · doi:10.1006/jnth.2000.2541
[27] Gaál, I.; Remete, L., Integral bases and monogenity of composite fields, Exp. Math., 28, 2, 209-222 (2019) · Zbl 1490.11106 · doi:10.1080/10586458.2017.1382404
[28] Gaál, I.; Remete, L., Power integral bases in cubic and quartic extensions of real quadratic fields, Acta Scientiarum Mathematicarum, 85, 34, 413-429 (2019) · Zbl 1449.11104 · doi:10.14232/actasm-018-080-z
[29] Gaál, I.; Remete, L.; Szabó, T., Calculating power integral bases by using relative power integral bases, Funct. Approx. Comment. Math., 54, 2, 141-149 (2016) · Zbl 1395.11121 · doi:10.7169/facm/2016.54.2.1
[30] Gaál, I.; Szabó, T., Relative power integral bases in infinite families of quartic extensions of quadratic fields, J. Algebra Number Theory Appl., 29, 1, 31-43 (2013) · Zbl 1335.11094
[31] Gouvêa, Fernando Q., Webster, J.: Dedekind on higher congruences and index divisors, 1871 and 1878. 2021. arXiv:2107.08905 [math.NT]
[32] Gouvêa, F.Q., Webster, J.: Kurt Hensel on common inessential discriminant divisors, 1894(2021). arXiv:2108.05327 [math.HO]
[33] Györy, K., Sur les polynômes à coefficients entiers et de discriminant donné, Acta Arith., 23, 419-426 (1973) · Zbl 0269.12001 · doi:10.4064/aa-23-4-419-426
[34] Györy, K., Sur les polynômes à coefficients entiers et de discriminant donné. II, Publ. Math. Debrecen, 21, 125-144 (1974) · Zbl 0303.12001 · doi:10.5486/PMD.1974.21.1-2.18
[35] Györy, K., Sur les polynômes à coefficients entiers et de discriminant donné. III, Publ. Math. Debrecen, 23, 1-2, 141-165 (1976) · Zbl 0354.10041
[36] Györy, K., Sur les polynômes à coefficients entiers et de discriminant donné. IV, Publ. Math. Debrecen, 25, 155-167 (1978) · Zbl 0405.12003
[37] Györy, K., Sur les polynômes à coefficients entiers et de discriminant donné. V, Acta Math. Acad. Sci. Hungar., 32, 175-190 (1978) · Zbl 0402.10053
[38] Györy, K.: Corps de nombres algébriques d’anneau d’entiers monogène. In: Séminaire Delange-Pisot-Poitou, 20e année: (1978/1979). Théörie des nombres, Fasc. 2 (French). Secrétariat Math., Paris, Exp. No. 26, 7 (1980) · Zbl 0433.12001
[39] Györy, K., On discriminants and indices of integers of an algebraic number field, J. Reine Angew. Math., 324, 114-126 (1981) · Zbl 0446.12006 · doi:10.1515/crll.1981.324.114
[40] Hall, M., Indices in cubic fields, Bull. Am. Math. Soc., 43, 2, 104-108 (1937) · Zbl 0016.00803 · doi:10.1090/S0002-9904-1937-06503-7
[41] Hansen, VL, Embedding finite covirng spaces into trivial bundles, Math. Ann., 236, 239-243 (1978) · Zbl 0368.55001 · doi:10.1007/BF01351369
[42] Hansen, VL, Polynomial covering spaces and homomorphisms into the braid groups, Pac. J. Math., 81, 399-410 (1979) · Zbl 0413.57001 · doi:10.2140/pjm.1979.81.399
[43] Hansen, VL, Coverings defined by Weierstrass polynomials, J. Reine Angew. Math., 314, 29-39 (1980) · Zbl 0421.57001
[44] Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer, New York, pp. xvi+496 (1977) · Zbl 0367.14001
[45] Hensel, K., Arithmetische Untersuchungen über die gemeinsamen ausserwesentlichen Discriminantentheiler einer Gattung, J. Reine Angew. Math., 113, 128-160 (1894) · JFM 25.0136.01 · doi:10.1515/crll.1894.113.128
[46] Herr, L.: If the normalization is affine, is it affine? (if quasiaffine). MathOver ow. version. Accessed 30 Nov 2020. https://mathoverflow.net/q/377840
[47] Herr, L.: When is a twisted form coming from a torsor trivial? (answer). MathOver ow. version. Accessed 6 Sept 2020. https://mathoverflow.net/q/370972
[48] Ji, L., et al. Weil restriction for schemes and beyond (2017). http://www-personal.umich.edu/ stevmatt/weil_restriction.pdf
[49] König, J., A note on families of monogenic number fields, Kodai Math. J., 41, 2, 456-464 (2018) · Zbl 1406.11109 · doi:10.2996/kmj/1530496853
[50] Lin, VJ, Algebroid functions and holomorphic elements of homotopy groups of a complex manifold, Soviet Math. Dokl., 12, 1608-1612 (1971) · Zbl 0261.46054
[51] Lønsted, K., On the embedding of coverings into 1-dimensional bundles, J. Reine Angew. Math., 317, 88-101 (1980) · Zbl 0424.55012
[52] Mayer, KH; Schwartzenberger, RLE, Non-embedding theorems for Y-spaces, Proc. Camb. Philos. Soc., 63, 601-612 (1967) · Zbl 0204.23605 · doi:10.1017/S030500410004158X
[53] Narkiewicz, W: Elementary and analytic theory of algebraic numbers, 3rd. Springer Monographs in Mathematics. Springer, Berlin, pp. xii+708 (2004). doi:10.1007/978-3-662-07001-7 · Zbl 1159.11039
[54] Narkiewicz, W.: The story of algebraic numbers in the first half of the 20th century: from Hilbert to Tate. Springer Monographs in Mathematics. Springer, Cham, pp. xi+443 (2018). ISBN: 978-3-030-03753-6; 978-3-030-03754-3 · Zbl 1416.11003
[55] Olsson, MC, Hom-stacks and restriction of scalars, Duke Math. J., 134, 1, 139-164 (2006) · Zbl 1114.14002 · doi:10.1215/S0012-7094-06-13414-2
[56] Ottaviani, G.: Introduction to the hyperdeterminant and to the rank of multidimensional matrices. ArXiv e-prints, arXiv:1301.0472 [math.AG] (2013) · Zbl 1276.14078
[57] Pleasants, PAB, The number of generators of the integers of a number field, Mathematika, 21, 160-167 (1974) · Zbl 0328.12008 · doi:10.1112/S0025579300008548
[58] Poonen, B., The moduli space of commutative algebras of finite rank, J. Eur. Math. Soc., 10, 131 (2006) · Zbl 1151.14011 · doi:10.4171/JEMS/131
[59] Prill, D., Primitive holomorphic maps of curves, Manuscripta Math., 32, 59-80 (1980) · Zbl 0444.14021 · doi:10.1007/BF01298182
[60] Röhrl, H.: Question 13 of Appendix, Proceedings of the Conference on Complex Analysis, Minneapolis 1964. Berlin (1965)
[61] Rydh, D.: Families of cycles and the Chow scheme (2008). (PhD dissertation, KTH). http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-4813
[62] Samsky, SR, Finite covers and modules of functions, Math. Ann., 236, 117-123 (1978) · Zbl 0381.32013 · doi:10.1007/BF01351385
[63] Serre, J.-P.: Local fields. Vol. 67. Graduate Texts in Mathematics. Translated from the French by Marvin Jay Greenberg, pp. viii+241. Springer, New York. ISBN: 0-387-90424-7 (1979) · Zbl 0423.12016
[64] Siad, A.: Monogenic fields with odd class number Part I. Odd degree (2020). arXiv: 2011.08834 [math.NT]
[65] Siad, A.: Monogenic fields with odd class number Part II. Even degree (2020). arXiv: 2011.08842 [math.NT]
[66] Stout, EL, Extensions of rings of holomorphic functions, Math. Ann., 196, 275-280 (1972) · Zbl 0233.32014 · doi:10.1007/BF01428217
[67] Stutz, J., Primitive elements for modules over O(Y ), Duke Math. J., 41, 329-333 (1974) · Zbl 0285.32012 · doi:10.1215/S0012-7094-74-04137-4
[68] Swaminathan, A.: Most Integral Odd-Degree Binary Forms Fail to Properly Represent a Square (2019). doi:10.48550/ARXIV.1910.12409
[69] Spearman, BK; Yang, Q.; Yoo, J., Minimal indices of pure cubic fields, Arch. Math., 106, 1, 35-40 (2016) · Zbl 1333.11099 · doi:10.1007/s00013-015-0812-z
[70] The LMFDB Collaboration. The L-functions and modular forms database. http://www.lmfdb.org. Accessed 15 July 2021 (2021)
[71] The Sage Developers. SageMath, the Sage Mathematics Software System (Version 8.7). https://www.sagemath.org (2019)
[72] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu (2020)
[73] Vojta, P.: Jets via Hasse-Schmidt Derivations. arXiv Mathematics e-prints (2004). arXiv:math/0407113 [math.AG]
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.