様々な重さ半整数の保型形式に関連する数論

设M是奇数且无平方的。Kohnen通过定义Kohnen加新空间，并证明了Hecke同构于2k，Level M的新形式空间，给出了4m级半整权(k+1/2)型的新理论。在全空间S_k+1/2(Γ_0(4m))和S_2k(Γ_0(2m))之间。我们寻找映射Hecke Isom的4m级半整权形式的子空间。到2m级的整体重量新形式上。我们构造了这样一个子空间，称它为4m级的负空间。为了构造负空间，我们需要从下面描述的真Hecke代数中得到某些算子.设G_p是由某个2-上循环定义的SL_2(Q_P)的双覆盖，K_0(P)是覆盖映射下Γ_0(P)Z_P的逆象，γ是真中心特征标.我们研究了G_p的Hecke代数H(G_p//K_0(P)，γ)，它对应于K_0(P)和γ，并给出了它在γ为二次时的生成元和关系的表示。这推广了Loke-Savin对H(G_2//K_0(4)，γ)的描述。利用4m级k+1/2权自同构型与S_k+1/2(Γ_0(4m))之间的Waldspurger同构，将H(G_p//K_0(P)，γ)中的元素转化为S_k+1/2(Γ_0(4m))上的经典算子。我们得到了具有特征值p和-1的经典算子Q_p和一个对合W_p，设Q‘_p是Q_p与W_p的共轭，负空间是Q_p和Q’_p的公共-1特征空间，所有素数p除以2m。这与我们之前在积分权重设置中的结果类似。

Let M be odd and square-free. Kohnen gave a newform theory of half-integral weight (k+1/2) forms of level 4M bydefining 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_punder the covering map and γ be a genuine central character. We study the Hecke algebra H(G_p//K_0(p), γ) of G_pcorresponding 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.