Newforms of Half-integral Weight: The Minus Space Counterpart

Baruch, Ehud Moshe ; Purkait, Soma

Canadian journal of mathematics, Tome 72 (2020) no. 2, pp. 326-372

Résumé

We study genuine local Hecke algebras of the Iwahori type of the double cover of $\operatorname{SL}_{2}(\mathbb{Q}_{p})$ and translate the generators and relations to classical operators on the space $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$, $M$ odd and square-free. In [9] Manickam, Ramakrishnan, and Vasudevan defined the new space of $S_{k+1/2}(\unicode[STIX]{x1D6E4}_{0}(4M))$ that maps Hecke isomorphically onto the space of newforms of $S_{2k}(\unicode[STIX]{x1D6E4}_{0}(2M))$. We characterize this newspace as a common $-1$-eigenspace of a certain pair of conjugate operators that come from local Hecke algebras. We use the classical Hecke operators and relations that we obtain to give a new proof of the results in [9] and to prove our characterization result.

Détail

DOI : 10.4153/S0008414X19000233

Mots-clés : Hecke algebra, half-integral weight form, Niwa isomorphism, Kohnen plus space

Bibliographie

[1] Atkin, A. O. L. and Lehner, J., Hecke operators on 𝛤(m). Math. Ann. 185(1970), 134–160. https://doi.org/10.1007/BF01359701 Google Scholar

[2] Baruch, E. M. and Purkait, S., Hecke algebras, new vectors and newforms on 𝛤(m). Math. Zeit. 287(2017), 705–733. https://doi.org/10.1007/s00209-017-1842-y Google Scholar

[3] Baruch, E. M. and Purkait, S., Newforms of half-integral weight: the minus space of S (𝛤(8M)). Israel J. Math. 232(2019), 41–73. https://doi.org/10.1007/s11856-019-1873-7 Google Scholar

[4] Gelbart, S., Weil’s representation and the spectrum of the metaplectic group. Lecture Notes in Mathematics, 530, Springer-Verlag, Berlin, 1976. Google Scholar

[5] Kohnen, W., Modular forms of half-integral weight on 𝛤(4). Math. Ann. 248(1980), 249–266. https://doi.org/10.1007/BF01420529 Google Scholar

[6] Kohnen, W., Newforms of half-integral weight. J. Reine Angew. Math. 333(1982), 32–72. https://doi.org/10.1515/crll.1982.333.32 Google Scholar

[7] Kumar, N. and Purkait, S., A note on the Fourier coefficients of half-integral weight modular forms. Arch. Math. (Basel) 102(2014), no. 4, 369–378. https://doi.org/10.1007/s00013-014-0622-8 Google Scholar

[8] Loke, H. Y. and Savin, G., Representations of the two-fold central extension of [[()[]mml:mo lspace="1em" rspace="0em"[]()]]SL[[()[]/mml:mo[]()]](ℚ). Pacific J. Math. 247(2010), 435–454. https://doi.org/10.2140/pjm.2010.247.435 Google Scholar

[9] Manickam, M., Ramakrishnan, B., and Vasudevan, T., On the theory of newforms of half-integral weight. J. Number Theory 34(1990), 210–224. https://doi.org/10.1016/0022-314X(90)90151-G Google Scholar

[10] Niwa, S., On Shimura’s trace formula. Nagoya Math. J. 66(1977), 183–202. Google Scholar

[11] Purkait, S., On Shimura’s decomposition. Int. J. Number Theory 9(2013), 1431–1445. https://doi.org/10.1142/S179304211350036X Google Scholar

[12] Purkait, S., Hecke operators in half-integral weight. J. Théor. Nombres Bordeaux 26(2014), 233–251. Google Scholar

[13] Savin, G., On unramified representations of covering groups. J. Reine Angew. Math. 566(2004), 111–134. https://doi.org/10.1515/crll.2004.001 Google Scholar

[14] Shimura, G., On modular forms of half integral weight. Ann. of Math. 97(1973), 440–481. https://doi.org/10.2307/1970831 Google Scholar

[15] Shimura, G., The critical values of certain zeta functions associated with modular forms of half-integral weight. J. Math. Soc. Japan 33(1981), 649–672. https://doi.org/10.2969/jmsj/03340649 Google Scholar

[16] Ueda, M., On twisting operators and newforms of half-integral weight. Nagoya Math J. 131(1993), 135–205. https://doi.org/10.1017/S002776300000458X Google Scholar

[17] Ueda, M. and Yamana, S., On newforms for Kohnen plus spaces. Math. Z. 264(2010), 1–13. https://doi.org/10.1007/s00209-008-0449-8 Google Scholar

[18] 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