Abstract
This is a historical survey of the Hecke operators and their action on the theta functions and the theta series of integral quadratic forms. Bibliography: 33 titles.
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References
E. Hecke,Analytische Arithmetik der Positiven Quadratischen Formen, Danske Vidensk. Selsk. Math.-Fys. Meddel. XIII, 12, Copenhagen (1940) (Math. Werke, pp. 789–918).
B. Schoeneberg, “Das Verhalten von mehrfachen Thetareihen bei Modulsubstitutionen,”Math. Ann.,116, 511–523 (1939).
E. Hecke, “Über Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I, II,”Math. Ann.,114, 1–28, 316–351 (1937) (Math. Werke pp. 644–707).
H. Petersson, “Konstruktion der sämtlichen Lösungen einer Riemannschen Funktionalgleichung durch Dirichletreihen mit Eulerscher Produktentwicklung. I, II, III,”Math. Ann.,116, 401–412 (1939),117, 39–64, 277–300 (1940/41).
K. Wohlfart, “Über Operatoren Heckescher Art bei Modulformen reeler Dimension,”Math. Nachr.,16, 233–256 (1957).
G. Shimura, “On modular forms of half integral weight,”Ann. Math. 97, 440–481 (1973).
A. N. Andrianov and G. N. Maloletkin, “The behavior of theta series of genusn of indefinite quadratic forms under modular substitutions”Tr. Mat. Inst. Akad. Nauk SSSR,148, 5–15 (1978).
M. Sugawara, “On the transformation theory of Siegel's modular group of then-th degree,”Proc. Imp. Acad. Jpn.,13, 335–336 (1937).
M. Sugawara, “An invariant property of Siegel's modular functions,”Proc. Imp. Acad. Jpn.,14, 1–3 (1938).
H. Maass, “Die Primzahlen in der Theorie der Siegelschen Modulfunktionen,”Math. Ann.,124, 87–122 (1951).
A. N. Andrianov,Quadratic Forms and Hecke Operators (Grundlehren Math. Wiss. 286), Springer-Verlag, Berlin-Heidelberg (1987).
A. N. Andrianov and V. G. Zhuravlev,Modular Forms and Hecke Operators, Nauka, Moscow (1990) (The English translation is to appear in U. S. A. by AMS).
A. N. Andrianov, “Integral representations of quadratic forms by quadratic forms. Multiplicative properties,” in:Proc. Intern. Congress of Mathematicians, Warsaw (1983), Vol. 1, PWN-Polish Sci. Publ., Warsaw, North-Holland, Amsterdam, New York, Oxford (1984), pp. 465–474.
E. Freitag, “Die Invarianz gewisser von Thetareihen erzeugter Vektorräume unter Heckeoperatoren,”Math. Z.,156, 141–155 (1977).
E. Freitag, “Berichtigung zu der Arbeit “Die Invarianz gewisser von Thetareihen erzeugter Vektorräume unter Heckeoperatoren”,”Math. Z.,168, 289–290 (1979).
A. N. Andrianov, “The multiplicative arithmetic of Siegel modular forms,”Uspekhi Mat. Nauk,34, No. 1, 67–135 (1979).
E. Freitag, “Eine Bemerkung zu Andrianovs explizirten Formeln für die Wirkung der Heckeoperatoren auf Thetareihen,” in: E. B. Christoffel, Birkhäuser, Basel (1981), pp. 336–351.
V. G. Zhuravlev, “Representations of Hecke rings of theta series of odd quadratic forms,” in:All-Union Conference “Number Theory and Applications” [in Russian], Tbilisi (1985), pp. 81–82.
C. L. Siegel, “On the theory of indefinite quadratic forms,”Ann. Math.,45, 577–622 (1944).
C. L. Siegel,Lectures on Quadratic Forms (Notes by K. G. Ramanathan), Tata Inst. Fund. Res., Bombay (1955–56) (Reissued 1967).
C. F. Gauss,Disquisitiones Arithmeticae, Fleischer, Lipsiae, Leipzig (1801) (Werke, Bd I).
Yu. V. Linnik, “Quaternions and Caley numbers; some applications of quaternion arithmetic,”Usp. Mat. Nauk,4, No. 5, 49–98 (1949).
A. N. Andrianov, “The action of the Hecke operators on nonhomogeneous theta series,”Mat. Sb.,131, 275–293 (1986).
A. N. Andrianov, “Calculation of coefficients in the formulas for transformations of inhomogeneous theta series under the action of the Hecke operators,”Zap. Nauchn. Semin. LOMI,160, 9–15 (1987).
A. N. Andrianov, “Multiplicative properties of solutions of quadratic Diophantine problems,”Algebra Analiz,2, No. 1, 3–46 (1990).
N. Andrianov, “Composition of solutions of quadratic Diophantine equations,”Usp. Mat. Nauk.,46, No. 2, 3–40 (1991).
A. N. Andrianov, “Queen's lectures on arithmetical composition of quadratic forms,”Queen's Papers Pure Appl. Math.,92 (1992).
A. N. Andrianov, “Factorizations of integral representations of binary quadratic forms,”Algebra Analiz,5, No. 1, 3–29 (1993).
E. Freitag, “Singular modular forms and theta relations,”Lect. Notes Math.,1487 (1991).
A. N. Andrianov, “Symmetries of harmonic theta functions of integral quadratic forms,”Usp. Mat. Nauk,50, No. 4, 3–44 (1995).
A. N. Andrianov, “Spherical theta series,”Mat. Sb.,134, 291–305 (1987).
E. Freitag, “Die Wirkung von Heckeoperatoren auf Thetareihen mit harmonischen Koeffizienten,”Math. Ann.,258 419–440 (1982).
V. G. Zhuravlev, “Generalized Eichler-Brandt matrices, the Hecke operators, and vector-valued theta series,”Algebra Analiz,5, No. 3 (1993).
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Translated fromZapiski Nauchnykh Serninarov POMI, Vol. 226, 1996, pp. 5–13.
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Andrianov, A.N. On an advice of Yu. V. Linnik, Hecke operators, and theta functions. J Math Sci 89, 909–914 (1998). https://doi.org/10.1007/BF02358527
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DOI: https://doi.org/10.1007/BF02358527