Squarefree parts of polynomial values
David Krumm1
1 Department of Mathematics Colby College Waterville, ME 04901, USA
Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 3, pp. 699-724.

Given a separable nonconstant polynomial f(x) with integer coefficients, we consider the set S consisting of the squarefree parts of all the rational values of f(x), and study its behavior modulo primes. Fixing a prime p, we determine necessary and sufficient conditions for S to contain an element divisible by p. We conjecture that if p is large enough, then S contains infinitely many representatives from every nonzero residue class modulo p. The conjecture is proved by elementary means assuming f(x) has degree 1 or 2. If f(x) has degree 3, or if it has degree 4 and has a rational root, the conjecture is shown to follow from the parity conjecture for elliptic curves. For polynomials of arbitrary degree, a local analogue of the conjecture is proved using standard results from class field theory, and empirical evidence is given to support the global version of the conjecture.

Étant donné un polynôme non constant séparable f(x) à coefficients entiers, nous considérons l’ensemble S constitué des parties sans facteurs carrés de toutes les valeurs rationnelles de f(x), et étudions son comportement modulo un nombre premier. Ayant fixé un nombre premier p, nous déterminons des conditions nécessaires et suffisantes pour que S contienne un élément divisible par p. Nous conjecturons que si p est suffisamment grand, alors S contient une infinité de représentants de chaque classe résiduelle non nulle modulo p. Nous prouvons cette conjecture quand f(x) est de degré 1 ou 2. Si f(x) est de degré 3, ou s’il est de degré 4 avec une racine rationnelle, la preuve de la conjecture utilise la conjecture de parité pour les courbes elliptiques. Pour les polynômes de degré arbitraire, un analogue local de la conjecture est prouvé en utilisant des résultats standard de la théorie des corps de classe. Des résultats numériques sont aussi inclus qui confirment la version globale de la conjecture.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.959
Classification : 11A07, 11R09, 11G05, 11G30
Mots-clés : Squarefree part, hyperelliptic curve, quadratic twist, congruent number

David Krumm 1

1 Department of Mathematics Colby College Waterville, ME 04901, USA 
@article{JTNB_2016__28_3_699_0,
     author = {David Krumm},
     title = {Squarefree parts of polynomial values},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {699--724},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {28},
     number = {3},
     year = {2016},
     doi = {10.5802/jtnb.959},
     zbl = {1415.11148},
     mrnumber = {3610693},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.959/}
}
TY  - JOUR
AU  - David Krumm
TI  - Squarefree parts of polynomial values
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2016
SP  - 699
EP  - 724
VL  - 28
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.959/
DO  - 10.5802/jtnb.959
LA  - en
ID  - JTNB_2016__28_3_699_0
ER  - 
%0 Journal Article
%A David Krumm
%T Squarefree parts of polynomial values
%J Journal de théorie des nombres de Bordeaux
%D 2016
%P 699-724
%V 28
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.959/
%R 10.5802/jtnb.959
%G en
%F JTNB_2016__28_3_699_0
David Krumm. Squarefree parts of polynomial values. Journal de théorie des nombres de Bordeaux, Tome 28 (2016) no. 3, pp. 699-724. doi : 10.5802/jtnb.959. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.959/

[1] S. Y. An, S. Y. Kim, D. C. Marshall, S. H. Marshall, W. G. McCallum & A. R. Perlis, « Jacobians of genus one curves », J. Number Theory 90 (2001), no. 2, p. 304-315. | DOI | MR | Zbl

[2] M. F. Atiyah & I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969, ix+128 pages. | DOI | Zbl

[3] D. Berend & Y. Bilu, « Polynomials with roots modulo every integer », Proc. Amer. Math. Soc. 124 (1996), no. 6, p. 1663-1671. | DOI | Zbl

[4] W. Bosma, J. Cannon & C. Playoust, « The Magma algebra system. I. The user language », J. Symbolic Comput. 24 (1997), no. 3-4, p. 235-265, Computational algebra and number theory (London, 1993). | DOI | MR | Zbl

[5] J. W. S. Cassels, Lectures on elliptic curves, London Mathematical Society Student Texts, vol. 24, Cambridge University Press, Cambridge, 1991, vi+137 pages. | Zbl

[6] D. A. Cox, Primes of the form x2+ny2, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1989, Fermat, class field theory and complex multiplication, xiv+351 pages. | DOI | Zbl

[7] J. R. Doyle, X. Faber & D. Krumm, « Preperiodic points for quadratic polynomials over quadratic fields », New York J. Math. 20 (2014), p. 507-605. | Zbl

[8] D. S. Dummit & R. M. Foote, Abstract algebra, third ed., John Wiley & Sons, Inc., Hoboken, NJ, 2004, xii+932 pages. | Zbl

[9] F. Gouvêa & B. Mazur, « The square-free sieve and the rank of elliptic curves », J. Amer. Math. Soc. 4 (1991), no. 1, p. 1-23. | DOI | MR | Zbl

[10] A. Granville, « Rational and integral points on quadratic twists of a given hyperelliptic curve », Int. Math. Res. Not. IMRN (2007), no. 8, p. Art. ID 027, 24. | DOI | Zbl

[11] M. Hindry & J. H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000, An introduction, xiv+558 pages. | DOI | Zbl

[12] J. W. P. Hirschfeld, G. Korchmáros & F. Torres, « The number of points on an algebraic curve over a finite field », in Surveys in combinatorics 2007, London Math. Soc. Lecture Note Ser., vol. 346, Cambridge Univ. Press, Cambridge, 2007, p. 175-200. | DOI | Zbl

[13] E. W. Howe, K. E. Lauter & J. Top, « Pointless curves of genus three and four », in Arithmetic, geometry and coding theory (AGCT 2003), Sémin. Congr., vol. 11, Soc. Math. France, Paris, 2005, p. 125-141. | Zbl

[14] K. Ireland & M. Rosen, A classical introduction to modern number theory, second ed., Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York, 1990, xiv+389 pages. | DOI | Zbl

[15] N. Koblitz, Introduction to elliptic curves and modular forms, second ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993, x+248 pages. | DOI | Zbl

[16] D. Krumm, « Quadratic points on modular curves », PhD Thesis, University of Georgia (USA), 2013.

[17] J. C. Lagarias & A. M. Odlyzko, « Effective versions of the Chebotarev density theorem », in Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, p. 409-464. | Zbl

[18] F. Lemmermeyer, Reciprocity laws, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000, From Euler to Eisenstein, xx+487 pages. | DOI | Zbl

[19] Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002, Translated from the French by Reinie Erné, Oxford Science Publications, xvi+576 pages. | Zbl

[20] D. Maisner & E. Nart, « Abelian surfaces over finite fields as Jacobians », Experiment. Math. 11 (2002), no. 3, p. 321-337, With an appendix by Everett W. Howe. | DOI | MR | Zbl

[21] P. Monsky, « Mock Heegner points and congruent numbers », Math. Z. 204 (1990), no. 1, p. 45-67. | DOI | MR | Zbl

[22] H. L. Montgomery & R. C. Vaughan, Multiplicative number theory. I. Classical theory, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007, xviii+552 pages. | Zbl

[23] J. Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften, vol. 322, Springer-Verlag, Berlin, 1999, xviii+571 pages. | DOI | Zbl

[24] B. Poonen, « Squarefree values of multivariable polynomials », Duke Math. J. 118 (2003), no. 2, p. 353-373. | DOI | MR | Zbl

[25] J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, no. 7, Springer-Verlag, New York-Heidelberg, 1973, viii+115 pages. | DOI | Zbl

[26] A. Silverberg, « Open questions in arithmetic algebraic geometry », in Arithmetic algebraic geometry (Park City, UT, 1999), IAS/Park City Math. Ser., vol. 9, Amer. Math. Soc., Providence, RI, 2001, p. 83-142. | DOI | Zbl

[27] J. H. Silverman, « Lower bounds for height functions », Duke Math. J. 51 (1984), no. 2, p. 395-403. | DOI | MR | Zbl

[28] —, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009, xx+513 pages. | DOI

[29] W. Stein et al., « Sage Mathematics Software (Version 6.2) », The Sage Development Team, 2014, http://www.sagemath.org.

[30] C. L. Stewart & J. Top, « On ranks of twists of elliptic curves and power-free values of binary forms », J. Amer. Math. Soc. 8 (1995), no. 4, p. 943-973. | DOI | MR | Zbl

[31] J. Top, « A remark on the rank of Jacobians of hyperelliptic curves over Q over certain elementary abelian 2-extensions », Tohoku Math. J. (2) 40 (1988), no. 4, p. 613-616. | DOI | MR | Zbl

Cité par Sources :