Abstract
In a series of recent papers, V.I. Arnold studied many questions concerning the statistics and dynamics of powers of elements in algebraic systems. In particular, on the basis of experimental data, he proposed an Euler-type congruence for the traces of powers of integer matrices as a conjecture. The proof of this conjecture was deduced from the author’s theorem (obtained at the end of 2004) on congruences for the traces of powers of elements in number fields. Recently, it turned out that there also exist other approaches to congruences for the traces of powers of integer matrices. In the present paper, the author’s results of 2004 are strengthened and a survey of their relations to number theory, theory of dynamical systems, combinatorics, and p-adic analysis is given. The main conclusion of this survey is that all approaches considered here ultimately reflect different points of view on a certain simple but important phenomenon in mathematics.
Similar content being viewed by others
References
Algebraic Number Theory, Ed. by J. W. C. Cassels and A. Fröhlich (Academic, London, 1967; Mir, Moscow, 1969).
V. I. Arnold, “Fermat-Euler Dynamical Systems and the Statistics of Arithmetics of Geometric Progressions,” Funkts. Anal. Prilozh. 37(1), 1–18 (2003) [Funct. Anal. Appl. 37, 1–15 (2003)].
V. I. Arnold, “The Topology of Algebra: Combinatorics of Squaring,” Funkts. Anal. Prilozh. 37(3), 20–35 (2003) [Funct. Anal. Appl. 37, 177–190 (2003)].
V. I. Arnold, Euler Groups and Arithmetic of Geometric Progressions (MTsNMO, Moscow, 2003) [in Russian].
V. I. Arnol’d, “Topology and Statistics of Formulae of Arithmetics,” Usp. Mat. Nauk 58(4), 3–28 (2003) [Russ. Math. Surv. 58, 637–664 (2003)].
V. I. Arnold, “Fermat Dynamics, Matrix Arithmetics, Finite Circles, and Finite Lobachevsky Planes,” Funkts. Anal. Prilozh. 38(1), 1–15 (2004) [Funct. Anal. Appl. 38, 1–13 (2004)].
V. I. Arnol’d, “The Matrix Euler-Fermat Theorem,” Izv. Ross. Akad. Nauk, Ser. Mat. 68(6), 61–70 (2004) [Izv. Math. 68, 1119–1128 (2004)].
V. I. Arnol’d, “Geometry and Dynamics of Galois Fields,” Usp. Mat. Nauk 59(6), 23–40 (2004) [Russ. Math. Surv. 59, 1029–1046 (2004)].
V. I. Arnold, “Number-Theoretical Turbulence in Fermat-Euler Arithmetics and Large Young Diagrams Geometry Statistics,” J. Math. Fluid Mech. 7(Suppl. 1), S4–S50 (2005).
V. I. Arnold, “Ergodic and Arithmetical Properties of Geometrical Progression’s Dynamics and of Its Orbits,” Moscow Math. J. 5(1), 5–22 (2005).
V. I. Arnold, “On the Matricial Version of Fermat-Euler Congruences,” Jpn. J. Math., Ser. 3, 1, 1–24 (2006).
I. K. Babenko and S. A. Bogatyi, “Lefschetz Numbers, Local Indices, and Periodic Points,” Dokl. Akad. Nauk SSSR 291(3), 521–524 (1986) [Sov. Math., Dokl. 34, 492–495 (1987)].
I. K. Babenko and S. A. Bogatyi, “The Behavior of the Index of Periodic Points under Iterations of a Mapping,” Izv. Akad. Nauk SSSR, Ser. Mat. 55(1), 3–31 (1991) [Math. USSR, Izv. 38, 1–26 (1992)].
S. A. Bogatyi, “The Number of Periodic Points of a Mapping of an Interval Grows Exponentially,” Soobshch. Akad. Nauk Gruz. SSR 121(1), 25–28 (1986).
S. A. Bogatyi, “Indexes of Iterations of Multi-valued Mappings,” C. R. Acad. Bulg. Sci. 41(2), 13–16 (1988).
Z. I. Borevich and I. R. Shafarevich, Number Theory (Nauka, Moscow, 1985) [in Russian].
N. Bourbaki, Eléments de mathématique. Algèbre commutative. Chapitre 8: Dimension. Chapitre 9: Anneaux locaux noethériens complets (Masson, Paris, 1983).
R. F. Brown, “Wecken Properties for Manifolds,” in Nielsen Theory and Dynamical Systems, Ed. by C. K. McCord (Am. Math. Soc., Providence, RI, 1993), Contemp. Math. 152, pp. 9–21.
Chong-Yun Chao, “Generalizations of Theorems of Wilson, Fermat and Euler,” J. Number Theory 15(1), 95–114 (1982).
Cohomologies p-adiques et applications arithmétiques (II), Ed. by P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, and M. Rapoport (Soc. Math. France, Paris, 2002), Astérisque 279.
Cohomologies p-adiques et applications arithmétiques (III), Ed. by P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, and M. Rapoport (Soc. Math. France, Paris, 2004), Astérisque 295.
L. E. Dickson, History of the Theory of Numbers (Chelsea, New York, 1971), Vol. 1.
A. Dold, Lectures on Algebraic Topology (Springer, Berlin, 1972; Mir, Moscow, 1976).
A. Dold, “Fixed Point Indices of Iterated Maps,” Invent. Math. 74, 419–435 (1983).
A. Fel’shtyn and R. Hill, “Dynamical Zeta Functions, Nielsen Theory and Reidemeister Torsion,” in Nielsen Theory and Dynamical Systems, Ed. by C. K. McCord (Am. Math. Soc., Providence, RI, 1993), Contemp. Math. 152, pp. 43–68.
J. Franks and D. Fried, “The Lefschetz Function of a Point,” in Topological Fixed Point Theory and Applications (Springer, Berlin, 1989), Lect. Notes Math. 1411, pp. 83–87.
P. R. Heath, R. Piccinini, and C. You, “Nielsen-Type Numbers for Periodic Points. I,” in Topological Fixed Point Theory and Applications (Springer, Berlin, 1989), Lect. Notes Math. 1411, pp. 88–106.
H. Hopf, “Über die algebraische Anzahl von Fixpunkten,” Math. Z. 29, 493–524 (1929).
W. Jänichen, “Über die Verallgemeinerung einer Gaussschen Formel aus der Theorie der höheren Kongruenzen,” Sitzungsber. Berlin. Math. Ges. 20, 23–29 (1921).
N. Koblitz, p-Adic Numbers, p-adic Analysis, and Zeta-Functions (Springer, New York, 1977; Mir, Moscow, 1982).
K. Komiya, “Congruences for Fixed Point Indices of Equivariant Maps and Iterated Maps,” in Topological Fixed Point Theory and Applications (Springer, Berlin, 1989), Lect. Notes Math. 1411, pp. 130–136.
K. Komiya, “Fixed Point Indices of Equivariant Maps and Möbius Inversion,” Invent. Math. 91, 129–135 (1988).
M. A. Krasnosel’skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis (Nauka, Moscow, 1975; Springer, Berlin, 1984).
S. Lang, Algebraic Numbers (Addison-Wesley, Reading, MA, 1964; Mir, Moscow, 1966).
J. Petersen, Tiddskr. Mat. (3), 2, 2, 64–65 (1872).
T. Schönemann, “Grundzüge einer allgemeinen Theorie der höhern Congruenzen, deren Modul eine reelle Primzahl ist,” J. Reine Angew. Math. 31, 269–325 (1846).
I. Schur, “Arithmetische Eigenschaften der Potenzsummen einer algebraischen Gleichung,” Compos. Math. 4, 432–444 (1937).
J.-P. Serre, Corps locaux (Hermann, Paris, 1962).
M. Shub and D. Sullivan, “A Remark on the Lefschetz Fixed Point Formula for Differentiable Maps,” Topology 13, 189–191 (1974).
S. Smale, “Differentiable Dynamical Systems,” Bull. Am. Math. Soc. 73, 747–817 (1967).
C. J. Smyth, “A Coloring Proof of a Generalization of Fermat’s Little Theorem,” Am. Math. Mon. 93(6), 469–471 (1986).
E. H. Spanier, Algebraic Topology (McGraw-Hill, New York, 1966; Mir, Moscow, 1971).
T. Szele, “Une généralisation de la congruence de Fermat,” Mat. Tidsskr. B, 57–59 (1948).
A. Thue, “Ein Kombinatorischer Beweis eines Satzes von Fermat,” Christiana Vid. Selsk. Skr. I. Mat. Nat. Kl., No. 3 (1910); repr. in Selected Mathematical Papers of Axel Thue (Universitelsforlaget, Oslo, 1977).
H. Ulrich, Fixed Point Theory of Parametrized Equivariant Maps (Springer, Berlin, 1988), Lect. Notes Math. 1343.
E. B. Vinberg, “Fermat’s Little Theorem and Its Generalizations,” Mat. Prosveshch., Ser. 3, No. 12, 1–11 (2008).
I. M. Vinogradov, Foundations of Number Theory (Nauka, Moscow, 1981) [in Russian].
J. Westlund, Proc. Indiana Acad. Sci., 78–79 (1902).
P. Wong, “Equivariant Nielsen Fixed Point Theory for G-Maps,” Pac. J. Math. 150(1), 179–200 (1991).
P. Wong, “Equivariant Nielsen Numbers,” Pac. J. Math. 159(1), 153–175 (1993).
P. P. Zabreiko and M. A. Krasnosel’skii, “Iterations of Operators and Fixed Points,” Dokl. Akad. Nauk SSSR 196(5), 1006–1009 (1971) [Sov. Math., Dokl. 12, 294–298 (1971)].
P. P. Zabreiko and M. A. Krasnosel’skii, “The Rotation of Vector Fields with Compositions and Iterations of Operators,” Vestn. Yaroslav. Univ., No. 12, 23–37 (1975).
A. V. Zarelua, “On Matrix Analogs of Fermat’s Little Theorem,” Mat. Zametki 79(6), 838–853 (2006) [Math. Notes 79, 783–796 (2006)].
O. Zariski and P. Samuel, Commutative Algebra (D. Van Nostrand, Princeton, NJ, 1958, 1960; Inostrannaya Literatura, Moscow, 1963), Vols. 1, 2.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.V. Zarelua, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 85–105.
Rights and permissions
About this article
Cite this article
Zarelua, A.V. On congruences for the traces of powers of some matrices. Proc. Steklov Inst. Math. 263, 78–98 (2008). https://doi.org/10.1134/S008154380804007X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S008154380804007X