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

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.

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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.596
@article{JTNB_2007__19_2_433_0,
     author = {Harald A. Helfgott},
     title = {Power-free values, large deviations, and integer points on irrational curves},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {433--472},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {2},
     year = {2007},
     doi = {10.5802/jtnb.596},
     zbl = {pre05302784},
     mrnumber = {2394896},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.596/}
}
TY  - JOUR
AU  - Harald A. Helfgott
TI  - Power-free values, large deviations, and integer points on irrational curves
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2007
DA  - 2007///
SP  - 433
EP  - 472
VL  - 19
IS  - 2
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.596/
UR  - https://zbmath.org/?q=an%3Apre05302784
UR  - https://www.ams.org/mathscinet-getitem?mr=2394896
UR  - https://doi.org/10.5802/jtnb.596
DO  - 10.5802/jtnb.596
LA  - en
ID  - JTNB_2007__19_2_433_0
ER  - 
Harald A. Helfgott. Power-free values, large deviations, and integer points on irrational curves. Journal de Théorie des Nombres de Bordeaux, Tome 19 (2007) no. 2, pp. 433-472. doi : 10.5802/jtnb.596. https://jtnb.centre-mersenne.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 1175278 | Zbl 0759.60007

[2] E. Bombieri, Le grand crible dans la théorie analytique des nombres. Astérisque 18, SMF, 1974. | MR 891718 | Zbl 0292.10035

[3] E. Bombieri, The Mordell conjecture revisited. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), no. 4, 615–640. | Numdam | MR 1093712 | Zbl 0722.14010

[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 1016893 | Zbl 0718.11048

[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 717589 | Zbl 0529.14015

[6] J. H. Conway, A. Hulpke, and J. McKay, On transitive permutation groups. LMS J. Comput. Math. 1 (1998), 1–8. | MR 1635715 | Zbl 0920.20001

[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 920369 | Zbl 0634.52002

[8] P. Corvaja and U. Zannier, On the number of integral points on algebraic curves. J. Reine Angew. Math. 565 (2003), 27–42. | MR 2024644 | Zbl pre02021326

[9] H. Davenport, Multiplicative number theory. Markham, Chicago, 1967. | MR 217022 | Zbl 0159.06303

[10] A. Dembo and O. Zeitouni, Large deviations techniques and applications. 2nd ed., Springer–Verlag, New York, 1998. | MR 1619036 | Zbl 0896.60013

[11] P. Erdős, Arithmetical properties of polynomials. J. London Math. Soc. 28 (1953), 416–425. | MR 56635 | Zbl 0051.27703

[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 2374

[13] T. Estermann, Einige Sätze über quadratfreie Zahlen. Math. Ann. 105 (1931), 653–662. | MR 1512732

[14] The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.3. , 2002.

[15] A. Granville, ABC allows us to count squarefrees. Internat. Math. Res. Notices 1998, no. 19, 991–1009. | MR 1654759 | Zbl 0924.11018

[16] G. Greaves, Power-free values of binary forms. Quart. J. Math. Oxford 43(2) (1992), 45–65. | MR 1150469 | Zbl 0768.11034

[17] G. Greaves, Sieves in number theory. Springer–Verlag, Berlin, 2001. | MR 1836967 | Zbl 1003.11044

[18] B. H. Gross, Local heights on curves. In G. Cornell, J. H. Silverman, eds., Arithmetic Geometry, Springer–Verlag, New York, 1986. | MR 861983 | Zbl 0605.14027

[19] H. Halberstam and K. F. Roth, On the gaps between consecutive k-free integers. J. London Math. Soc. 26 (1951), 268–273. | MR 43120 | Zbl 0043.04901

[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 2099831 | Zbl 1057.11043

[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 2220098 | Zbl 1127.14029

[24] M. Hindry and J. H. Silverman, Diophantine geometry. Springer–Verlag, New York, 2000. | MR 1745599 | Zbl 0948.11023

[25] F. den Hollander, Large deviations. AMS, Providence, RI, 2000. | MR 1739680 | Zbl 0949.60001

[26] C. Hooley, Applications of sieve methods to the theory of numbers. Cambridge University Press, Cambridge, 1976. | MR 404173 | Zbl 0327.10044

[27] C. Hooley, On power-free numbers and polynomials. I. J. Reine Angew. Math. 293/294 (1977), 67–85. | MR 457388 | Zbl 0354.10038

[28] C. Hooley, On power-free numbers and polynomials. II. J. Reine Angew. Math. 295 (1977), 1–21. | MR 457389 | Zbl 0354.10038

[29] M. Huxley and M. Nair, Power free values of polynomials, III. Proc. London Math. Soc. (3) 41 (1980), no. 1, 66–82. | MR 579717 | Zbl 0435.10026

[30] H. Iwaniec and E. Kowalski, Analytic number theory. AMS Colloquium Publications, v. 53, AMS, Providence, RI, 2004. | MR 2061214 | Zbl 1059.11001

[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 514023 | Zbl 0407.52005

[32] S. Lang, Algebraic number theory. 2nd ed., Springer-Verlag, New York, 1994. | MR 1282723 | Zbl 0811.11001

[33] S. Lang, Fundamentals of diophantine geometry. Springer–Verlag, New York, 1983. | MR 715605 | Zbl 0528.14013

[34] S. Lang, Number Theory III, Diophantine geometry. Springer–Verlag, New York, 1991. | MR 1112552 | Zbl 0744.14012

[35] V. I. Levenshtein, Universal bounds for codes and designs. Handbook of coding theory, North-Holland, Amsterdam, Vol I., 499–648. | MR 1667942 | Zbl 0983.94056

[36] M. Nair, Power free values of polynomials II. Proc. London Math. Soc. (3) 38 (1979), no. 2, 353–368. | MR 531167 | Zbl 0404.10020

[37] P. M. Neumann, A lemma that is not Burnside’s. Math. Sci. 4 (1979), 133–141. | Zbl 0409.20001

[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 419382 | Zbl 0329.10035

[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 1465369 | Zbl 0888.11023

[40] A. Parson and J. Tull, Asymptotic behavior of multiplicative functions. J. Number Theory 10 (1978), no. 4, 395–420. | MR 515052 | Zbl 0392.10038

[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 174533 | Zbl 0141.04305

[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 116378 | Zbl 0112.10106

[44] E. F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve. Math. Ann. 310 (1998), 447–471. | MR 1612262 | Zbl 0889.11021

[45] W. R. Scott, Group Theory. 2nd ed., Dover, New York, 1987. | MR 896269 | Zbl 0641.20001

[46] J.-P. Serre, Lectures on the Mordell-Weil theorem. 3rd ed., Vieweg, Braunschweig, 1997. | MR 1757192 | Zbl 0676.14005

[47] J. H. Silverman, Arithmetic distance functions and height functions in diophantine geometry. Math. Ann. 279 (1987), 193–216. | MR 919501 | Zbl 0607.14013

[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 2169047 | Zbl 1086.11032

Cité par Sources :