Abstract
We compute generators and relations for a certain 2-adic Hecke algebra of level 8 associated with the double cover of SL2 and a 2-adic Hecke algebra of level 4 associated with PGL2. We show that these two Hecke algebras are isomorphic as expected from the Shimura correspondence. We use the 2-adic generators to define classical Hecke operators on the space of holomorphic modular forms of weight k + 1/2 and level 8M where M is odd and square-free. Using these operators and our previous results on half-integral weight forms of level 4M we define a subspace of the space of half-integral weight forms as a common -1 eigenspace of certain Hecke operators. Using the relations and a result of Ueda we show that this subspace, which we call the minus space, is isomorphic as a Hecke module under the Ueda correspondence to the space of new forms of weight 2k and level 4M. We observe that the forms in the minus space satisfy a Fourier coefficient condition that gives the complement of the plus space but does not define the minus space.
Similar content being viewed by others
References
A. O. L. Atkin and J. Lehner, Hecke operators on Γ0(m), Mathematische Annalen 185 (1970), 134–160.
E. M. Baruch and S. Purkait, Hecke algebras, new vectors and newforms on Γ0(m), Mathematische Zeitschrift 287 (2017), 705–733.
E. M. Baruch and S. Purkait, Newforms of half-integral weight: the minus space counterpart, Canadian Journal of Mathematics, to appear, arXiv:1609.06481v2.
S. Gelbart, Weil’s Representation and the Spectrum of the Metaplectic Group, Lecture Notes in Mathematics, Vol. 530, Springer, Berlin–New York, 1976.
W. Kohnen, Modular forms of half-integral weight on Γ0(4), Mathematische Annalen 248 (1980), 249–266.
W. Kohnen, Newforms of half-integral weight, Journal für die Reine und Angewandte Mathematik 333 (1982), 32–72.
H. Y. Loke and G. Savin, Representations of the two-fold central extension of SL2(Q2), Pacific Journal of Mathematics 247 (2010), 435–454.
M. Manickam, B. Ramakrishnan and T. Vasudevan, On the theory of newforms of halfintegral weight, Journal of Number Theory 34 (1990), 210–224.
M. Manickam, J. Meher and B. Ramakrishnan, Theory of newforms of half-integral weight, Pacific Journal of Mathematics 274 (2015), 125–139.
S. Purkait, On Shimura’s decomposition, International Journal of Number Theory 9 (2013), 1431–1445.
G. Shimura, On modular forms of half integral weight, Annals of Mathematics 97 (1973), 440–481.
M. Ueda, The decomposition of the spaces of cusp forms of half-integral weight and trace formula of Hecke operators, Journal of Mathematics of Kyoto University 28–3 (1988), 505–555.
M. Ueda, On twisting operators and newforms of half-integral weight, Nagoya Mathematical Journal 131 (1993), 135–205.
M. Ueda, Newforms of half-integral weight in the case of level 2m.
M. Ueda and S. Yamana, On newforms for Kohnen plus spaces, Mathematische Zeitshcrift 264 (2010), 1–13.
J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demientier, Journal de Mathématiques Pures et Appliquées. Neuvième Série 60 (1981), 375–484.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Baruch, E.M., Purkait, S. Newforms of half-integral weight: the minus space of Sk+1/2(Γ0(8M)). Isr. J. Math. 232, 41–73 (2019). https://doi.org/10.1007/s11856-019-1873-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-019-1873-7