Power-free values, large deviations, and integer points on irrational curves
Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 433-472.

Let f[x] be a polynomial of degree d3 without roots of multiplicity d or (d-1). Erdős conjectured that, if f satisfies the necessary local conditions, then f(p) is free of (d-1)th powers for infinitely many primes p. This is proved here for all f with sufficiently high entropy.

The proof serves to demonstrate two innovations: a strong repulsion principle for integer points on curves of positive genus, and a number-theoretical analogue of Sanov’s theorem from the theory of large deviations.

Soit f[x] un polynôme de degré d3 sans racines de multiplicité d ou (d-1). Erdős a conjecturé que si f satisfait les conditions locales nécessaires alors f(p) est sans facteurs puissances (d-1) èmes pour une infinité de nombres premiers p. On prouve cela pour toutes les fonctions f dont l’entropie est assez grande.

On utilise dans la preuve un principe de répulsion pour les points entiers sur les courbes de genre positif et un analogue arithmétique du théorème de Sanov issu de la théorie des grandes déviations.

DOI: 10.5802/jtnb.596
Helfgott, Harald A. 1

1 Département de mathématiques et de statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada
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Helfgott, Harald A. Power-free values, large deviations, and integer points on irrational curves. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 2, pp. 433-472. doi : 10.5802/jtnb.596. http://archive.numdam.org/articles/10.5802/jtnb.596/

[1] R. Arratia and S. Tavare, The cycle structure of random permutations. Ann. Probab. 20 (1992), 1567–1591. | MR | Zbl

[2] E. Bombieri, Le grand crible dans la théorie analytique des nombres. Astérisque 18, SMF, 1974. | Numdam | MR | Zbl

[3] E. Bombieri, The Mordell conjecture revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 615–640. | Numdam | MR | Zbl

[4] E. Bombieri and J. Pila, The number of integral points on arcs and ovals. Duke Math. J. 59 (1989), no. 2, 337–357. | MR | Zbl

[5] J. W. S. Cassels, The Mordell-Weil group of curves of genus 2. Arithmetic and Geometry, Vol. I, Birkäuser, Boston, 1983, 27–60. | MR | Zbl

[6] J. H. Conway, A. Hulpke, and J. McKay, On transitive permutation groups. LMS J. Comput. Math. 1 (1998), 1–8. | MR | Zbl

[7] J. H. Conway and N. J. A. Sloane, Sphere packings, lattices and groups. Grundlehren der Mathematischen Wissenschaften 290, Springer–Verlag, New York, 1988. | MR | Zbl

[8] P. Corvaja and U. Zannier, On the number of integral points on algebraic curves. J. Reine Angew. Math. 565 (2003), 27–42. | MR

[9] H. Davenport, Multiplicative number theory. Markham, Chicago, 1967. | MR | Zbl

[10] A. Dembo and O. Zeitouni, Large deviations techniques and applications. 2nd ed., Springer–Verlag, New York, 1998. | MR | Zbl

[11] P. Erdős, Arithmetical properties of polynomials. J. London Math. Soc. 28 (1953), 416–425. | MR | Zbl

[12] P. Erdős and M. Kac, The Gaussian law of errors in the theory of additive number theoretic functions. Amer. J. Math. 62 (1940), 738–742. | MR

[13] T. Estermann, Einige Sätze über quadratfreie Zahlen. Math. Ann. 105 (1931), 653–662. | MR

[14] The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.3. http://www.gap-system.org, 2002.

[15] A. Granville, ABC allows us to count squarefrees. Internat. Math. Res. Notices 1998, no. 19, 991–1009. | MR | Zbl

[16] G. Greaves, Power-free values of binary forms. Quart. J. Math. Oxford 43(2) (1992), 45–65. | MR | Zbl

[17] G. Greaves, Sieves in number theory. Springer–Verlag, Berlin, 2001. | MR | Zbl

[18] B. H. Gross, Local heights on curves. In G. Cornell, J. H. Silverman, eds., Arithmetic Geometry, Springer–Verlag, New York, 1986. | MR | Zbl

[19] H. Halberstam and K. F. Roth, On the gaps between consecutive k-free integers. J. London Math. Soc. 26 (1951), 268–273. | MR | Zbl

[20] R. Heath-Brown, Counting rational points on algebraic varieties. C.I.M.E. lecture notes, to appear.

[21] H. A. Helfgott, On the behaviour of root numbers in families of elliptic curves. Submitted, math.NT/0408141.

[22] H. A. Helfgott, On the square-free sieve. Acta Arith. 115 (2004), 349–402. | MR | Zbl

[23] H. A. Helfgott and A. Venkatesh, Integral points on elliptic curves and 3-torsion in class groups. To appear in J. Amer. Math. Soc. | MR | Zbl

[24] M. Hindry and J. H. Silverman, Diophantine geometry. Springer–Verlag, New York, 2000. | MR | Zbl

[25] F. den Hollander, Large deviations. AMS, Providence, RI, 2000. | MR | Zbl

[26] C. Hooley, Applications of sieve methods to the theory of numbers. Cambridge University Press, Cambridge, 1976. | MR | Zbl

[27] C. Hooley, On power-free numbers and polynomials. I. J. Reine Angew. Math. 293/294 (1977), 67–85. | MR | Zbl

[28] C. Hooley, On power-free numbers and polynomials. II. J. Reine Angew. Math. 295 (1977), 1–21. | MR | Zbl

[29] M. Huxley and M. Nair, Power free values of polynomials, III. Proc. London Math. Soc. (3) 41 (1980), no. 1, 66–82. | MR | Zbl

[30] H. Iwaniec and E. Kowalski, Analytic number theory. AMS Colloquium Publications, v. 53, AMS, Providence, RI, 2004. | MR | Zbl

[31] G. A. Kabatjanskii and V. I. Levenshtein, Bounds for packings on the sphere and in space (Russian). Problemy Peredači Informacii 14 (1978), no. 1, 3–25. | MR | Zbl

[32] S. Lang, Algebraic number theory. 2nd ed., Springer-Verlag, New York, 1994. | MR | Zbl

[33] S. Lang, Fundamentals of diophantine geometry. Springer–Verlag, New York, 1983. | MR | Zbl

[34] S. Lang, Number Theory III, Diophantine geometry. Springer–Verlag, New York, 1991. | MR | Zbl

[35] V. I. Levenshtein, Universal bounds for codes and designs. Handbook of coding theory, North-Holland, Amsterdam, Vol I., 499–648. | MR | Zbl

[36] M. Nair, Power free values of polynomials II. Proc. London Math. Soc. (3) 38 (1979), no. 2, 353–368. | MR | Zbl

[37] P. M. Neumann, A lemma that is not Burnside’s. Math. Sci. 4 (1979), 133–141. | Zbl

[38] K. K. Norton, On the number of restricted prime factors of an integer, I. Illinois J. Math. 20 (1976), no. 4, 681–705. | MR | Zbl

[39] B. Poonen and E. F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488 (1997), 141–188. | MR | Zbl

[40] A. Parson and J. Tull, Asymptotic behavior of multiplicative functions. J. Number Theory 10 (1978), no. 4, 395–420. | MR | Zbl

[41] K. Ramsay, personal communication.

[42] G. J. Rieger, Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positive Schranke, II. J. Reine Angew. Math. 217 (1965), 200–216. | MR | Zbl

[43] I. N. Sanov, On the probability of large deviations of random variables (in Russian). Mat. Sb. N. S. 42 (84) (1957), 11–44. English translation in: Select. Transl. Math. Statist. and Probability I (1961), 213–244. | MR | Zbl

[44] E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310 (1998), 447–471. | MR | Zbl

[45] W. R. Scott, Group Theory. 2nd ed., Dover, New York, 1987. | MR | Zbl

[46] J.-P. Serre, Lectures on the Mordell-Weil theorem. 3rd ed., Vieweg, Braunschweig, 1997. | MR | Zbl

[47] J. H. Silverman, Arithmetic distance functions and height functions in diophantine geometry. Math. Ann. 279 (1987), 193–216. | MR | Zbl

[48] P. Turán, Über einige Verallgemeinerungen eines Satzes von Hardy und Ramanujan. J. London Math. Soc. 11 (1936), 125–133.

[49] M. Young, Low-lying zeros of families of elliptic curves. J. Amer. Math. Soc. 19 (2006), 205–250. | MR | Zbl

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