On Congruences for the Traces of Powers of Some Matrices

A. V. Zarelua

Informatics and Automation, Geometry, topology, and mathematical physics. I, Tome 263 (2008), pp. 85-105.

Résumé

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.

Bibliographie

[1] Dzh. Kassels, A. Frëlikh (red.), Algebraicheskaya teoriya chisel, Mir, M., 1969 | MR

[2] Arnold V. I., “Dinamicheskaya sistema Ferma–Eilera i statistika arifmetiki geometricheskikh progressii”, Funkts. analiz i ego pril., 37:1 (2003), 1–18 | MR | Zbl

[3] Arnold V. I., “Topologiya algebry: kombinatorika operatsii vozvedeniya v kvadrat”, Funkts. analiz i ego pril., 37:3 (2003), 20–35 | MR | Zbl

[4] Arnold V. I., Gruppy Eilera i arifmetika geometricheskikh progressii, MTsNMO, M., 2003

[5] Arnold V. I., “Topologiya i statistika formul arifmetiki”, UMN, 58:4 (2003), 3–28 | MR | Zbl

[6] Arnold V. I., “Dinamika Ferma, arifmetika matrits, konechnaya okruzhnost i konechnaya ploskost Lobachevskogo”, Funkts. analiz i ego pril., 38:1 (2004), 1–15 | MR | Zbl

[7] Arnold V. I., “Matrichnaya teorema Eilera–Ferma”, Izv. RAN. Ser. mat., 68:6 (2004), 61–70 | MR | Zbl

[8] Arnold V. I., “Geometriya i dinamika polei Galua”, UMN, 59:6 (2004), 23–40 | MR | Zbl

[9] Arnold V. I., “Number-theoretical turbulence in Fermat–Euler arithmetics and large Young diagrams geometry statistics”, J. Math. Fluid Mech., 7, Suppl. 1 (2005), S4–S50 | DOI | MR | Zbl

[10] Arnold V. I., “Ergodic and arithmetical properties of geometrical progression's dynamics and of its orbits”, Moscow Math. J., 5:1 (2005), 5–22 | MR | Zbl

[11] Arnold V. I., “On the matricial version of Fermat–Euler congruences”, Japan. J. Math. Ser. 3, 1 (2006), 1–24 | DOI | MR | Zbl

[12] Babenko I. K., Bogatyi S. A., “Chisla Lefshetsa, lokalnye indeksy i periodicheskie tochki”, DAN SSSR, 291:3 (1986), 521–524 | MR

[13] Babenko I. K., Bogatyi S. A., “Povedenie indeksa periodicheskikh tochek pri iteratsiyakh otobrazheniya”, Izv. AN SSSR. Ser. mat., 55:1 (1991), 3–31 | MR | Zbl

[14] Bogatyi S. A., “Kolichestvo periodicheskikh tochek otobrazheniya otrezka rastet eksponentsialno”, Soobsch. AN Gruz. SSR, 121:1 (1986), 25–28 | MR | Zbl

[15] Bogatyi S. A., “Indeksy iteratsii mnogoznachnogo otobrazheniya”, Dokl. Bolg. AN, 41:2 (1988), 13–16 | MR | Zbl

[16] Borevich Z. I., Shafarevich I. R., Teoriya chisel, Nauka, M., 1985 | MR | Zbl

[17] Bourbaki N., Eléments de mathématique. Algèbre commutative. Chapitre 8: Dimension. Chapitre 9: Anneaux locaux noethériens complets, Masson, Paris, 1983 | MR | Zbl

[18] Brown R.F., “Wecken properties for manifolds iNielsen theory and dynamical systems”, Contemp. Math., 152, eds. C. K. McCord, Amer. Math. Soc., Providence, RI, 1993, 9–21 | MR | Zbl

[19] Chao Chong-Yun, “Generalizations of theorems of Wilson, Fermat and Euler”, J. Number Theory, 15:1 (1982), 95–114 | DOI | MR | Zbl

[20] P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, M. Rapoport (eds.), Cohomologies $p$-adiques et applications arithmétiques II, Astérisque, 279, Soc. math. France, Paris, 2002 | MR | Zbl

[21] P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, M. Rapoport (eds.), Cohomologies $p$-adiques et applications arithmétiques III, Astérisque, 295, Soc. math. France, Paris, 2004 | MR | Zbl

[22] Dickson L. E., History of the theory of numbers, V. 1, Chelsea, New York, 1971

[23] Dold A., Lektsii po algebraicheskoi topologii, Mir, M., 1976 | MR

[24] Dold A., “Fixed point indices of iterated maps”, Invent. math., 74 (1983), 419–435 | DOI | MR | Zbl

[25] Fel'shtyn A., Hill R., “Dynamical zeta functions, Nielsen theory and Reidemeister torsion”, Nielsen theory and dynamical systems, Contemp. Math., 152, eds. C. K. McCord, Amer. Math. Soc., Providence, RI, 1993, 43–68 | MR

[26] Franks J., Fried D., “The Lefschetz function of a point”, Topological fixed point theory and applications, Lect. Notes Math., 1411, Springer, Berlin, 1989, 83–87 | MR

[27] Heath P. R., Piccinini R., You C., “Nielsen-type numbers for periodic points. I”, Topological fixed point theory and applications, Lect. Notes Math., 1411, Springer, Berlin, 1989, 88–106 | MR

[28] Hopf H., “Über die algebraische Anzahl von Fixpunkten”, Math. Ztschr., 29 (1929), 493–524 | DOI | MR | Zbl

[29] Jänichen W., “Über die Verallgemeinerung einer Gaussschen Formel aus der Theorie der höheren Kongruenzen”, Sitzungsber. Berl. Math. Ges., 20 (1921), 23–29 | Zbl

[30] Koblits N., $p$-Adicheskie chisla, $p$-adicheskii analiz i dzeta-funktsii, Mir, M., 1982 | MR

[31] Komiya K., “Congruences for fixed point indices of equivariant maps and iterated maps”, Topological fixed point theory and applications, Lect. Notes Math., 1411, Springer, Berlin, 1989, 130–136 | MR

[32] Komiya K., “Fixed point indices of equivariant maps and Möbius inversion”, Invent. math., 91 (1988), 129–135 | DOI | MR | Zbl

[33] Krasnoselskii M. A., Zabreiko P. P., Geometricheskie metody nelineinogo analiza, Nauka, M., 1975 | MR

[34] Leng S., Algebraicheskie chisla, Mir, M., 1966 | MR

[35] Petersen J., “Foundations of envelope theory”, Tiddskr. mat. (3), 2 (1872), 64–65

[36] Schönemann T., “Grundzüge einer allgemeinen Theorie der höhern Congruenzen, deren Modul eine reelle Primzahl ist”, J. reine und angew. Math., 31 (1846), 269–325 | Zbl

[37] Schur I., “Arithmetische Eigenschaften der Potenzsummen einer algebraischen Gleichung”, Compos. Math., 4 (1937), 432–444 | MR | Zbl

[38] Serre J.-P., Corps locaux, Hermann, Paris, 1962 | MR | Zbl

[39] Shub M., Sullivan D., “A remark on the Lefschetz fixed point formula for differentiable maps”, Topology, 13 (1974), 189–191 | DOI | MR | Zbl

[40] Smale S., “Differentiable dynamical systems”, Bull. Amer. Math. Soc., 73 (1967), 747–817 | DOI | MR

[41] Smyth C. J., “A coloring proof of a generalization of Fermat's little theorem”, Amer. Math. Month., 93:6 (1986), 469–471 | DOI | MR | Zbl

[42] Spener E., Algebraicheskaya topologiya, Mir, M., 1971 | MR

[43] Szele T., “Une généralisation de la congruence de Fermat”, Mat. tidsskr. B, 1948 (1948), 57–59 | MR | Zbl

[44] Thue A., “Ein Kombinatorischer Beweis eines Satzes von Fermat”, Kr. Vid. Selsk. Skr. I Mat. Nat. Kl., 1910, no. 3; Selected mathematical papers of Axel Thue, Universitelsforlaget, Oslo, 1977

[45] Ulrich H., Fixed point theory of parametrized equivariant maps, Lect. Notes Math., 1343, Springer, Berlin, 1988 | MR | Zbl

[46] Vinberg E. B., “Malaya teorema Ferma i ee obobscheniya”, Mat. prosveschenie. Ser. 3, 2008, no. 12, 43–53

[47] Vinogradov I. M., Osnovy teorii chisel, Nauka, M., 1981 | MR

[48] Westlund J., Proc. Indiana Ac. Sci., 1902, 78–79

[49] Wong P., “Equivariant Nielsen fixed point theory for $G$-maps”, Pacif. J. Math., 150:1 (1991), 179–200 | MR | Zbl

[50] Wong P., “Equivariant Nielsen numbers”, Pacif. J. Math., 159:1 (1993), 153–175 | MR | Zbl

[51] Zabreiko P. P., Krasnoselskii M. A., “Iteratsii operatorov i nepodvizhnye tochki”, DAN SSSR, 196:5 (1971), 1006–1009 | MR | Zbl

[52] Zabreiko P. P., Krasnoselskii M. A., “Vraschenie vektornykh polei s superpozitsiyami i iteratsiyami operatorov”, Vestn. Yarosl. un-ta, 1975, no. 12, 23–37 | MR

[53] Zarelua A. V., “O matrichnykh analogakh maloi teoremy Ferma”, Mat. zametki, 79:6 (2006), 838–853 | MR | Zbl

[54] Zarisskii O., Samyuel P., Kommutativnaya algebra, T. 1, 2, Izd-vo inostr. lit., M., 1963