様々な重さ半整数の保型形式に関連する数論

设M是奇数且无平方数。Kohnen通过定义Kohnen+新空间并证明Hecke同构于权为2k的新形式空间，给出了4 M级半整权（k+1/2）形式的新形式理论。在丹羽，有一个Hecke isom。在全空间S_k+1/2（Γ_0（4 M））和S_2k（Γ_0（2 M））之间。我们寻找一个子空间的半整数重量形式的水平4 M映射Hecke isom。到2 M层的整体重量新形式上。我们构造了这样一个子空间，称之为4 M层的负空间。为了构造负空间，我们需要从下面描述的真Hecke代数中得到的某些算子，设G_p是SL_2（Q_p）的由某个2-上圈定义的双覆盖，K_0（p）是覆盖映射下的Γ_0（p）Z_p的逆像，γ是真中心特征标。本文研究了G_p中对应于K_0（p）和γ的Hecke代数H（G_p//K_0（p），γ），并给出了当γ是二次时H（G_p//K_0（p），γ）的生成元和关系的表示.这推广了Loke-Savin关于H（G_2//K_0（4），γ）的描述.本文利用权为k+1/2、水平为4 M的椭圆自守形式空间与S_k+1/2（Γ_0（4 M））之间的Waldspurger同构，将H（G_p//K_0（p），γ）中对每个素数p整除4 M的元素转化为S_k+1/2（Γ_0（4 M））上的经典算子。我们得到了具有特征值p和-1和对合W_p的经典算子Q_p，设Q '_p是Q_p与W_p的共轭，负空间是Q_p和Q'_p对所有素数p整除2 M的公共-1特征空间。这类似于我们以前在积分权重设置中的结果。

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.