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A congruence for the Fourier coefficients of a modular form and its application to quadratic forms

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Abstract

Let F(z)=∑ ∞ n=1 A(n)q n denote the unique weight 6 normalized cuspidal eigenform on Γ0(4). We prove that A(p)≡0,2,−1(mod 11) when p≠11 is a prime. We then use this congruence to give an application to the number of representations of an integer by quadratic form of level 4.

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References

  1. Andrianov, A.N., Zhuravlev, V.G.: Modular Forms and Hecke Operators. Math. Monographs, vol. 145. AMS, Providence (1995)

    MATH  Google Scholar 

  2. Conway, J.H., Sloan, N.J.A.: Sphere Packings, Lattices and Groups, 3rd edn. Grundlehren der Mathematischen Wissenschaften, vol. 290. Springer, New York (1999)

    MATH  Google Scholar 

  3. Deligne, P.: Formes modulaires et représentations ℓ-adiques. In: Séminaire Bourbaki, 1968/69, Exp. No. 355. Lecture Notes in Mathematics, vol. 179. Springer, Berlin (1971)

    Chapter  Google Scholar 

  4. Koblitz, N.: Introduction to Elliptic Curves and Modular Forms, 2nd edn. GTM, vol. 97. Springer, Berlin (1993)

    MATH  Google Scholar 

  5. Mazur, B.: An introduction to the deformation theory of Galois representations. In: Modular Forms and Fermat’s Last Theorem, pp. 243–311. Springer, Berlin (1997)

    Google Scholar 

  6. Ono, K.: The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series. AMS, Providence (2004)

    MATH  Google Scholar 

  7. Serre, J.-P.: Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15, 259–331 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Serre, J.-P.: Congruences et formes modulaires (d’apres H.P.F. Swinnerton-Dyer). In: Séminaire Bourbaki, 24 année (1971/72), Exp. No. 416. Lecture Notes in Mathematics, vol. 317, pp. 74–88. Springer, Berlin (1972)

    Google Scholar 

  9. Serre, J.-P.: Représentations Linéaires des Groupes Finis, 5th edn. Hermann, Paris (1998)

    MATH  Google Scholar 

  10. Swinnerton-Dyer, H.P.F.: On ℓ-adic representations and congruences for coefficients of modular forms. In: Modular Functions of One Variable, vol. 3. Lecture Notes in Mathematics, vol. 350, pp. 1–55. Springer, Berlin (1973)

    Chapter  Google Scholar 

  11. Weil, A.: Number Theory. Birkhäuser, Basel (1983)

    MATH  Google Scholar 

  12. Wilton, J.R.: Congruence properties of Ramanujan’s function τ(n). Proc. Lond. Math. Soc. 31, 1–10 (1930)

    Article  Google Scholar 

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  1. Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu, 702-701, Korea

    Hyunsuk Moon

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  1. Hyunsuk Moon
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Moon, H. A congruence for the Fourier coefficients of a modular form and its application to quadratic forms. Ramanujan J 16, 73–81 (2008). https://doi.org/10.1007/s11139-007-9097-6

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  • DOI: https://doi.org/10.1007/s11139-007-9097-6

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