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Ternary quadratic forms and half-integral weight modular forms

Part of: Forms and linear algebraic groups Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 December 2012

Alia Hamieh
Alia Hamieh*
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road Vancouver, British Columbia, V6T 1Z2, Canada (email: [email protected])

Abstract

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Let k be a positive integer such that k≡3 mod 4, and let N be a positive square-free integer. In this paper, we compute a basis for the two-dimensional subspace Sk/2(Γ0(4N),F) of half-integral weight modular forms associated, via the Shimura correspondence, to a newform F∈Sk−1(Γ0(N)), which satisfies . This is accomplished by using a result of Waldspurger, which allows one to produce a basis for the forms that correspond to a given F via local considerations, once a form in the Kohnen space has been determined.

MSC classification

Secondary: 11F37: Forms of half-integer weight; nonholomorphic modular forms 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 15 , May 2012 , pp. 418 - 435
Copyright
© The Author(s) 2012

References

[1]Baruch, E. and Mao, Z., ‘Central value of automorphic L-functions’, Geom. Funct. Anal. 17 (2007) 333–384.CrossRefGoogle Scholar
[2]Böcherer, S. and Schulze-Pillot, R., ‘Vector valued theta series and Waldspurger’s theorem’, Abh. Math. Semin. Univ. Hambg. 64 (1994) 211–233.CrossRefGoogle Scholar
[3]Flicker, Y., ‘Automorphic forms on covering groups of GL(2)’, Invent. Math. 57 (1980) 119–182.CrossRefGoogle Scholar
[4]Gross, B., ‘Heights and the special values of L-series’, Conference Proceedings, Canadian Mathematical Society 7 (American Mathematical Society, Providence, RI, 1987) 115–187.Google Scholar
[5]Koblitz, N., ‘Introduction to Elliptic curves and modular forms’ 2nd edn, Graduate Texts in Mathematics 97 (Springer, New York, NY, 1993).Google Scholar
[6]Kohnen, W., ‘Newforms of half-integral weight’, J. reine angew. Math. 333 (1982) 32–72.Google Scholar
[7]Pacetti, A. and Tornaria, G., ‘Shimura correspondence for level p 2 and the central values of L-series’, J. Number Theory 124 (2007) 396–414.CrossRefGoogle Scholar
[8]Pizer, A., ‘An algorithm for computing modular forms on Γ0(n)’, J. Algebra 64 (1980) 340–390.CrossRefGoogle Scholar
[9]Stein, W. A., and others, ‘Sage Mathematics Software (Version 4.7)’ (The Sage Development Team, 2011), http://www.sagemath.org.Google Scholar
[10]Shimura, G., ‘On modular forms of half-integral weight’, Ann. of Math. (2) 97 (1973) 440–481.CrossRefGoogle Scholar
[11]Shintani, T., ‘On construction of holomorphic cusp forms of half integral weight’, Nagoya Math J. 58 (1975) 83–126.CrossRefGoogle Scholar
[12]Sturm, J., ‘Theta series of weight ’, J. Number Theory 14 (1982) 353–361.CrossRefGoogle Scholar
[13]Waldspurger, J.-L., ‘Correspondance de Shimura’, J. Math. Pures Appl. (9) 59 (1980) 1–132.Google Scholar
[14]Waldspurger, J.-L., ‘Sur les coefficients de Fourier des formes modulaires de poids demi-entier’, J. Math. Pures Appl. (9) 60 (1981) 375–484.Google Scholar
[15]Waldspurger, J.-L., ‘Correspondance de Shimura et quaternions’, Forum Math. 3 (1991) 219–307.CrossRefGoogle Scholar