Counting co-cyclic lattices. (English) Zbl 1348.11056

Nguyen, Phong Q.; Shparlinski, Igor E.

SIAM J. Discrete Math. 30, No. 3, 1358-1370 (2016).

Let \(\mathcal{I}_n\) denote the set of all full-rank integer lattices \(L\) in \(\mathbb{Z}^n\). For a positive integer \(V\) one defines \(\mathcal{I}_{n,V}\) (resp. \(\mathcal{I}_{n,\leq V}\))\(=\{ L\in \mathcal{I}_n \,|\,[\mathbb{Z}^n:L]=V \text{(resp. }\leq V\))

Reviewer: Detlev Hoffmann (Dortmund)

MSC:

11H06	Lattices and convex bodies (number-theoretic aspects)
11N60	Distribution functions associated with additive and positive multiplicative functions

Keywords:

lattice; co-cyclic lattice; homogeneous congruence; multiplicative function

Citations:

Zbl 0172.06304

References:

[1]	M. Ajtai, Generating hard instances of lattice problems, in Proceedings of the 28th ACM Symposium on Theory of Computing, ACM, New York, 1996, pp. 99–108. · Zbl 0921.11071
[2]	H. Cohen and H. W. Lenstra, Jr., Heuristics on class groups of number fields, in Number Theory, Noordwijkerhout 1983, Lect. Notes in Math 1068, H. Jager, ed., Springer, Berlin, 1984, pp. 33–62.
[3]	J. Fulman and L. Goldstein, Stein’s method and the rank distribution of random matrices over finite fields, Ann. Probab., 43 (2015), pp. 1274–1314. · Zbl 1388.60024
[4]	T. Ekedahl, An infinite version of the Chinese remainder theorem, Comment. Math. Univ. St. Paul., 40 (1991), pp. 53–59. · Zbl 0749.11004
[5]	N. Gama, M. Izabachène, P. Q. Nguyen, and X. Xie, Structural lattice reduction: Generalized worst-case to average-case reductions and homomorphic cryptosystems, in Advances in Cryptology—EUROCRYPT 2016, Lect. Notes Comput. Sci. 9666, M. Fischlin and J.-S. Coron, eds., Springer, Berlin, 2016, pp. 528–558. · Zbl 1371.94635
[6]	E.-U. Gekeler, Statistics about elliptic curves over finite prime fields, Manuscr. Math., 127 (2008), pp. 55–67. · Zbl 1156.11024
[7]	H. Gillet and D. R. Grayson, Volumes of symmetric spaces via lattice points, Doc. Math., 11 (2006), pp. 425–447. · Zbl 1116.11028
[8]	G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, Oxford, UK, 1979. · Zbl 0423.10001
[9]	C. J. Hillar and D. L. Rhea, Automorphisms of finite Abelian groups, Amer. Math. Monthly, 114 (2007), pp. 917–923. · Zbl 1156.20046
[10]	C. Hooley, A note on square-free numbers in arithmetic progressions, Bull. London Math. Soc., 7 (1975), pp. 133–138. · Zbl 0308.10030
[11]	E. Landau, Über die Zahlentheoretische Function \(φ(n)\) und ihre Beziehung zum Goldbachschen Satz, Nachr. Königlichen Ges. Wiss. Göttingen, Math.-Phys. Klasse, Göttingen, 1900, pp. 177–186. · JFM 31.0179.01
[12]	J. Lengler, The Cohen-Lenstra Heuristic for Finite Abelian Groups, Ph.D. thesis, Universität des Saarlandes, Saarbrüücken, Germany, 2009.
[13]	J. Lengler, The Cohen-Lenstra heuristic: Methodology and results, J. Algebra, 323 (2010), pp. 2960–2976. · Zbl 1213.11167
[14]	H. L. Montgomery, Primes in arithmetic progressions, Mich. Math. J., 17 (1970), pp. 33–39. · Zbl 0209.34804
[15]	I. Pak, On Probability of Generating a Finite Group, preprint, Department of Mathematics, Yale University, New Haven, CT, 1999.
[16]	A. Paz and C.-P. Schnorr, Approximating integer lattices by lattices with cyclic factor groups, in Automata, Languages and Programming: Proceedings of the 14th International Colloquium (Karlsruhe, Federal Republic of Germany, July 13––17, 1987), Lect. Notes Comput. Sci. 267, T. Ottmann, ed., Springer, Berlin, 1987, pp. 386–393. · Zbl 0642.10031
[17]	K. Prachar, Über die kleinste quadratfreie Zahl einer arithmetischen Reihe, Monatsh. Math., 62 (1958), pp. 173–176. · Zbl 0083.03704
[18]	O. Robert and P. Sargos, Three-dimensional exponential sums with monomials, J. Reine Angew. Math., 591 (2006), pp. 1–20. · Zbl 1165.11067
[19]	W. M. Schmidt, Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height, Duke Math. J., 35 (1968), pp. 327–339. · Zbl 0172.06304
[20]	M. Trolin, Lattices with many cycles are dense, in STACS 2004: Proceedings of the 21st Annual Symposium on Theoretical Aspects of Computer Science (Montpellier, France, March 25–27, 2004), Lect. Notes Comput. Sci. 2996, V. Diekert and M. Habib, eds., Springer-Verlag, Berlin, 2004, pp. 370–381. · Zbl 1122.94405
[21]	Y. Wang and R. P. Stanley, The Smith Normal Form Distribution of a Random Integer Matrix, preprint, http://arxiv.org/abs/1506.00160arXiv:1506.00160 [math.CO], 2015.
[22]	D. R. Ward, Some series involving Euler’s function, J. London Math. Soc., 210 (1927), pp. 1–2. · JFM 53.0164.01