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On the negative Pell equation. (English) Zbl 1230.11136

It is an interesting question for which fundamental discriminants \(D > 0\) there is a solution to the negative Pell equation \(x^2 - dy^2 = -1\) (here \(d = D/4\) if \(d\) is even and \(d=D\) otherwise). Such a solution can only exist if \(D\) has no prime divisor \(p \equiv 3\) (mod 4), hence the number \( \mathcal{D}^-(X)\) of such discriminants \(D \leq X\) is \(O(X/\sqrt{\log X})\). Considering discriminants with \(k\) prime factors, Odoni showed the lower bound \(\mathcal{D}^{-}(X) \gg_k X (\log\log X)^k/\log X\) which was improved by the reviwer to \(\gg X (\log X)^{-.62}\). On the basis of a probablistic model, Stevenhagen conjectured \(\mathcal{D}^-(X) \sim \alpha X /\sqrt{\log X}\) with \(\alpha = .58\ldots\).
The authors make great progress on this conjecture by showing \[ .41 \frac{X}{\sqrt{\log X}} \leq \mathcal{D}^-(X) \leq .67\frac{X}{\sqrt{\log X}} \] for \(X\) sufficiently large (ineffective!).
The starting point is the well-known criterion that the negative Pell equation is solvable if and only if for each \(k \geq 2\) the \(2^k\)-rank of the class group equals the \(2^k\)-rank of the narrow class group of the underlying number field. The authors use this for \(k=2\) (4-rank) to deduce their bounds. To this end, the authors study power moments of the type \[ \sum_{{D \leq X,\atop p \mid D \Rightarrow p \not\equiv 3\, (4)}} 2^{k \cdot \text{rk}_4(C_D)} \] and variants thereof. The asymptotic evaluation of these quantities requires very sophisticated algebraic and analytic tools that are too complicated to be described here.

MSC:

11R29 Class numbers, class groups, discriminants
11R11 Quadratic extensions
11E16 General binary quadratic forms
Full Text: DOI

References:

[1] V. Blomer, On the negative Pell equation, 2006.
[2] J. Brüdern, Einf” uhrung in die Analytische Zahlentheorie, Berlin: Springer-Lehrbuch, 1995. · Zbl 0830.11001
[3] H. Cohen and H. W. Lenstra Jr., ”Heuristics on class groups of number fields,” in Number Theory, New York: Springer-Verlag, 1984, vol. 1068, pp. 33-62. · Zbl 0558.12002 · doi:10.1007/BFb0099440
[4] H. Cohn, A Classical Invitation to Algebraic Numbers and Class Fields, New York: Springer-Verlag, 1978. · Zbl 0395.12001
[5] L. Comtet, Advanced Combinatorics, enlarged ed., Dordrecht: D. Reidel Publishing Co., 1974. · Zbl 0283.05001
[6] H. Davenport, Multiplicative Number Theory, Second ed., New York: Springer-Verlag, 1980, vol. 74. · Zbl 0453.10002
[7] J. A. Dieudonné, La Géométrie des Groupes Classiques, New York: Springer-Verlag, 1971. · Zbl 0221.20056
[8] L. P. G. Dirichlet, Vorlesungen über Zahlentheorie, New York: Chelsea Publishing Co., 1968. · JFM 03.0063.01
[9] T. Estermann, Introduction to Modern Prime Number Theory, Cambridge: Cambridge, at the Univ. Press, 1952, vol. 41. · Zbl 0049.03103
[10] &. Fouvry and J. Klüners, ”Cohen-Lenstra heuristics of quadratic number fields,” in Algorithmic Number Theory, New York: Springer-Verlag, 2006, vol. 4076, pp. 40-55. · Zbl 1143.11352 · doi:10.1007/11792086
[11] &. Fouvry and J. Klüners, ”The parity of the period of the continued fraction of \(\sqrtd\),” PLMS, vol. 101, pp. 337-391, 2010. · Zbl 1244.11092 · doi:10.1112/plms/pdp057
[12] &. Fouvry and J. Klüners, ”On the 4-rank of class groups of quadratic number fields,” Invent. Math., vol. 167, iss. 3, pp. 455-513, 2007. · Zbl 1126.11062 · doi:10.1007/s00222-006-0021-2
[13] J. Friedlander and H. Iwaniec, ”The polynomial \(X^2+Y^4\) captures its primes,” Ann. of Math., vol. 148, iss. 3, pp. 945-1040, 1998. · Zbl 0926.11068 · doi:10.2307/121034
[14] F. Gerth III, ”The \(4\)-class ranks of quadratic fields,” Invent. Math., vol. 77, iss. 3, pp. 489-515, 1984. · Zbl 0533.12004 · doi:10.1007/BF01388835
[15] L. J. Goldstein, ”A generalization of the Siegel-Walfisz theorem,” Trans. Amer. Math. Soc., vol. 149, pp. 417-429, 1970. · Zbl 0201.05701 · doi:10.2307/1995404
[16] G. H. Hardy and S. Ramanujan, ”The normal number of prime factors of a number \(n\),” Quart. J. Math., vol. 48, pp. 76-92, 1920. · JFM 46.0262.03
[17] H. Hasse, Number Theory, New York: Springer-Verlag, 1980, vol. 229. · Zbl 0423.12002
[18] D. R. Heath-Brown, ”The size of Selmer groups for the congruent number problem. II,” Invent. Math., vol. 118, iss. 2, pp. 331-370, 1994. · Zbl 0815.11032 · doi:10.1007/BF01231536
[19] D. R. Heath-Brown, ”A mean value estimate for real character sums,” Acta Arith., vol. 72, iss. 3, pp. 235-275, 1995. · Zbl 0828.11040
[20] D. R. Heath-Brown, ”Kummer’s conjecture for cubic Gauss sums,” Israel J. Math., vol. 120, iss. , part A, pp. 97-124, 2000. · Zbl 0989.11042
[21] E. Hecke, ”Eine neue Art von Zetafunktionen und ihre Beziehungen zur Verteilung der Primzahlen,” Math. Z., vol. 6, iss. 1-2, pp. 11-51, 1920. · JFM 47.0152.01 · doi:10.1007/BF01202991
[22] E. Hecke, Lectures on the Theory of Algebraic Numbers, New York: Springer-Verlag, 1981, vol. 77. · Zbl 0504.12001
[23] H. Heilbronn, ”On the averages of some arithmetical functions of two variables,” Mathematika, vol. 5, pp. 1-7, 1958. · Zbl 0125.02604 · doi:10.1112/S0025579300001273
[24] C. Hooley, ”On the Pellian equation and the class number of indefinite binary quadratic forms,” J. Reine Angew. Math., vol. 353, pp. 98-131, 1984. · Zbl 0539.10019 · doi:10.1515/crll.1984.353.98
[25] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Second ed., New York: Springer-Verlag, 1990, vol. 84. · Zbl 0712.11001
[26] H. Iwaniec and E. Kowalski, Analytic Number Theory, Providence, RI: Amer. Math. Soc., 2004, vol. 53. · Zbl 1059.11001
[27] G. J. Janusz, Algebraic Number Fields, Second ed., Providence, RI: Amer. Math. Soc., 1996, vol. 7. · Zbl 0854.11001
[28] M. Karoubi and T. Lambre, ”Sur la \(K\)-théorie du foncteur norme,” J. Algebra, vol. 321, iss. 10, pp. 2754-2781, 2009. · Zbl 1178.19003 · doi:10.1016/j.jalgebra.2008.09.044
[29] F. Lemmermeyer, The \(4\)-class group of real quadratic number fields. · Zbl 0634.12008
[30] S. Louboutin, ”Groupes des classes d’idéaux triviaux,” Acta Arith., vol. 54, iss. 1, pp. 61-74, 1989. · Zbl 0634.12008
[31] T. Mitsui, ”Generalized prime number theorem,” Jap. J. Math., vol. 26, pp. 1-42, 1956. · Zbl 0126.27503
[32] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Second ed., New York: Springer-Verlag, 1990. · Zbl 0717.11045
[33] L. Rédei and H. Reichardt, ”Die Anzahl der durch \(4\) teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers,” J. Reine Angew. Math., vol. 170, pp. 69-74, 1933. · Zbl 0009.29401
[34] L. Rédei, ”Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlk” orper,” J. Reine Angew. Math., vol. 171, pp. 55-60, 1934. · Zbl 0009.05101 · doi:10.1515/crll.1934.171.55
[35] L. Rédei, ”Eine obere Schranke der Anzahl der durch vier teilbaren invarianten der absoluten Klassengruppe im quadratischen Zahlk” orper,” J. Reine Angew. Math., vol. 171, pp. 61-64, 1934. · Zbl 0010.33801
[36] L. Rédei, ”Über die Grundeinheit und die durch 8 teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper,” J. Reine Angew. Math., vol. 171, pp. 131-148, 1934. · Zbl 0010.33802
[37] G. J. Rieger, ”Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke. II,” J. Reine Angew. Math., vol. 217, pp. 200-216, 1965. · Zbl 0141.04305 · doi:10.1515/crll.1965.217.200
[38] J. Serre, Local Fields, New York: Springer-Verlag, 1979. · Zbl 0423.12016
[39] A. Scholz, ”Über die Lösbarkeit der Gleichung \(t^2-Du^2=-4\),” Math. Z., vol. 39, iss. 1, pp. 95-111, 1935. · Zbl 0009.29402 · doi:10.1007/BF01201346
[40] P. Shiu, ”A Brun-Titchmarsh theorem for multiplicative functions,” J. Reine Angew. Math., vol. 313, pp. 161-170, 1980. · Zbl 0412.10030 · doi:10.1515/crll.1980.313.161
[41] P. Stevenhagen, ”The number of real quadratic fields having units of negative norm,” Experiment. Math., vol. 2, iss. 2, pp. 121-136, 1993. · Zbl 0792.11041 · doi:10.1080/10586458.1993.10504272
[42] A. Weil, Number Theory : An Approach Through History, From Hammurapi to Legendre, Boston, MA: Birkhäuser, 1984. · Zbl 0531.10001 · doi:10.1007/978-0-8176-4571-7
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