| 研究実績の概要 |
Let M be odd and square-free. Kohnen gave a newform theory of half-integral weight (k+1/2) forms of level 4M by
defining Kohnen plus new space and proving Hecke isomorphism to the space of newforms of weight 2k, level M. By Niwa,
there is a Hecke isom. between the full space S_k+1/2(Γ_0(4M)) and S_2k(Γ_0(2M)). We look for a subspace of half-integral weight forms of level 4M that maps Hecke isom. onto the integral weight newforms of level 2M. We construct such a subspace and call it the minus space of level 4M. In order to construct the minus space we need certain operators that we obtain from the genuine Hecke algebra described below.
Let G_p be the double cover of SL_2(Q_p) defined by a certain 2-cocycle, K_0(p) be the inverse image of Γ_0(p)Z_p
under the covering map and γ be a genuine central character. We study the Hecke algebra H(G_p//K_0(p), γ) of G_p
corresponding to K_0(p) and γ and give a presentation of it in terms of generators and relations when γ is quadratic. This generalizes Loke-Savin's description of H(G_2//K_0(4), γ). We use Waldspurger's isomorphism between the space of adelic automorphic forms of weight k+1/2, level 4M and S_k+1/2(Γ_0(4M)) to translate the elements in H(G_p//K_0(p), γ) for each prime p dividing 4M to classical operators on S_k+1/2(Γ_0(4M)). We obtain classical operators Q_p with eigenvalues p and -1 and an involution W_p. Let Q’_p be the conjugate of Q_p by W_p. The minus space is the common -1 eigenspace of Q_p and Q'_p for all primes p dividing 2M. This is analogous to our previous results in the integral weight setting.
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