Applications of the theory of modular forms to number theory

DOI10.1007/BF01261429zbMath0446.10021OpenAlexW1976660811MaRDI QIDQ1145715

Oleg M. Fomenko

Publication date: 1980

Published in: Journal of Soviet Mathematics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf01261429

zbMATH Keywords

surveyBibliographyarithmetic applications of modular functionsarithmetic questionstheory of modular forms

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) (11E45) Research exposition (monographs, survey articles) pertaining to number theory (11-02) Modular and automorphic functions (11F03) Congruences for modular and (p)-adic modular forms (11F33) Special values of automorphic (L)-series, periods of automorphic forms, cohomology, modular symbols (11F67) Holomorphic modular forms of integral weight (11F11) Automorphic forms, one variable (11F12)

Related Items (1)

Algebraic number fields

Cites Work

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• Quadratische Formen und Modulfunktionen
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• A Congruence Satisfied by the Theta-Constant ϑ 3
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• Decomposition Into Four Integral Squares in the Fields of 2 ½ and 3½
• Calculation of Class Numbers by Decomposition into Three Integral Squares in the Fields of 2 ½ and 3 ½
• Anwendung Jacobischer Thetafunktionen auf die Potenzreste
• Multiplicative functions and operators of Hecke type
• On the representation of a number as the sum of any number of squares, and in particular of twenty
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• Magnitude of the Fourier coefficients of automorphic forms of negative dimension
• Über Quadratische Charaktersummen
• Congruences for the Coefficients of the Modular Invariant $j(\tau)$.
• Note on Ramanujan's Function $\tau(n)$.
• Darstellungen von Korrespondenzen algebraischer Funktionenkörper und ihre Spuren.
• On the Fourier coefficients of triangle functions
• Variations sur un thème de Siegel et Hecke
• Zur Frage der Produktentwicklung höherer Thetafunktionen
• Products related to Gauss sums.
• Quaternary even positive definite quadratic forms of prime discriminant
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• The Fourier expansion of Epstein's zeta function for totally real algebraic number fields and some consequences for Dedekind's zeta function
• Calculation and Applications of Epstein Zeta Functions
• On some p-adic properties of the eichler-selberg trace formula
• Modular forms of half integral weight and the integral of certain theta-functions
• Fourier Coefficients of Eisenstein Series of One Epstein Series of One Complex Variable for the Special Linear Group
• Interwining Operators and Automorphic Forms for the Metaplectic Group
• A look back at Ramanujan's Notebooks
• On the Holomorphy of Certain Dirichlet Series
• Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type
• Multiplicative Congruence Properties and Density Problems for p(n)
• A correspondence between quaternary quadratic forms
• MacDonald Identities and Euclidean Lie Algebras
• On construction of holomorphic cusp forms of half integral weight
• Diophantine equations and modular forms
• Automorphic forms and algebraic extensions of number fields
• Dedekind Sums and a Paper of G. H. Hardy
• Dedekind's η-function and the cohomology of infinite dimensional Lie algebras
• Arithmetic of quaternary quadratic forms
• Formes modulaires de poids $1$
• On Complex Quadratic Fields with Class-Number Two
• Generalized Kloosterman Sums and the Fourier Coefficients of Cusp Forms
• Automorphic Forms on Adele Groups. (AM-83)
• Representations of quadratic forms and their application to Selberg’s zeta functions
• On a function analogous to log η(τ)
• An Ω‐result for the coefficients of cusp forms
• The theory of Hecke integrals
• Die Berechnung von Zetafunktionen mit Vorzeichencharakter an der Stelle 1
• A relation between automorphic forms on GL (2) and GL (3)
• Über einfache periodische Kettenbrüche und Vermutungen von P. Chowla und S. Chowla
• The reciprocity theorem for Dedekind-Rademacher sums
• Some formulas for the Riemann zeta function at odd integer argument resulting from Fourier expansions of the Epstein zeta function
• Calculation of the Ramanujan τ-Dirichlet Series
• On Fourier coefficients of certain cusp forms
• Some elliptic function identities
• Some Identities on the Character Sum Containing x(x - 1)(x - λ)
• A Note on Hecke Operators and Theta-Series
• On Ramanujan's formula for values of Riemann zeta-function at positive odd integers
• Eichler Maps and Hyperbolic Fourier Expansion
• Dedekind-Rademacher sums and their reciprocity formula.
• On the Trace of Hecke Operators for Certain Modular Groups
• On the relation between the positive definite quadratic forms with the same representation numbers
• Problems in the theory of automorphic forms to Salomon Bochner in gratitude
• Weber's class invariants
• Analytic Continuation of the Series ∑(m + nz) -s
• Über Bernoullische Zahlen in reell-quadratischen Zahlkörpern
• О представлении чисел положительными тернарными диагональными квадратичными формами, I
• An elementary proof of p(11m+6)≡0 (mod 11)
• Über Anwendungen höherer Dedekindscher Summen auf die Struktur elementar-arithmetischer Klasseninvarianten reell-quadratischer Zahlkörper.
• An Average value for Ramanujan's τ-Function
• A Result in the Theory of Weierstrass Elliptic Functions
• Asymptotic Formulae for the Coefficients of a Class of Modular Forms.
• Analytic Continuation of Eisenstein Series
• On the values of the Epstein zeta function
• On certain identities between the traces of Hecke operators
• On the coefficients of the $2^n$-th transformation polynomial for j(ω)
• Generalized Dedekind Eta-Functions and Generalized Dedekind Sums
• Ramanujan-Identitäten zur Reziproken der Jacobischen Funktion .. .
• Dedekind Sums for a Fuchsian Group, I
• Évaluation d'une somme arithmétique. Remarques
• On the coefficients of transformation polynomials for the modular function
• A simplification of the formula for L(1,χ) where χ is a totally imaginary Dirichlet character of a real quadratic field
• Cusp Forms Arising from Hilbert's Modular Functions for the Field of 3 ½
• A numerical study of Dedekind's cubic class number formula
• Divisibility Properties of the Fourier Coefficients of the Modular Invariant j(τ)
• Further Congruence Properties of the Fourier Coefficients of the Modular Invariant j(τ)
• Proof of Ramanujan's Partition Congruence for the Modulus 11 3
• Remarks on Some Modular Identities
• The Scalar Product of Modular Forms
• The Coefficients of Certain Infinite Products
• NOTE ON SOME PARTITION FORMULAE
• Some Properties of Partitions
• On Horocyclic Groups
• Über die Quaternionen in der Theorie der elliptischen Modulfunktionen.
• EVEN QUADRATIC FORMS WITH DETERMINANT UNITY
• The Ramanujan Identities Under Modular Substitutions
• Ramanujan Identities Involving the Partition Function for the Moduli 11 a
• Properties of the Coefficients of the Modular Invariant J(τ)
• Trends in research: The analytic number theory
• An Identity for the Coefficients of Certain Modular Forms
• Approximation to Irrationals by Classes of Rational Numbers
• Hyperelliptic modular curves
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