受阻双曲Kac-Moody分母的强制阴影
Forced Shadows of an Obstructed Hyperbolic Kac-Moody Denominator
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中文总结 AI 辅助
本文研究四元数代数B6的反射壁数据与Borcherds分母，证明受阻分母的阴影是Hecke本征形式，结合有限性界确定对应D值，还给出Petersson几何的刚性结论及相关范数性质。
中文摘要 AI 辅助
四元数代数B6的四阶在符号为(3,2)的格上带有四个反射壁数据；其中三个可积分到Borcherds分母，一个受阻。该失败的分母作为弱调和Maass形式保留，我们证明其阴影是新形式6.4.a.a线上的Hecke本征形式，具有零扭曲分量。机制为不变性选择：受阻泛函在判别式等距群下不变，该群在S_{5/2}中的不变量是一维的；在四元数判别式10和22处验证的相同机制，将那里的阴影置于10.4.a.a和22.4.a.c。在权重1/2层，我们证明一个确定性定理：当受阻空间消失时，典范形式无条件且唯一存在，结合Bruinier-Ehlen-Freitag的有限性界与有限计算，这恰好发生在D∈{6,10,22}时，这些是亏格为零的紧Shimura曲线，其极大阶三元组是反射的，具有秩为3、4、4的外尔腔。奇偶性将该层限制在奇通道；在受阻取向上，截面层在一个明确的40元素取向轨迹外受阻，产生双重阴影：权重3/2处的CM（36.2.a.a）和权重5/2处的新形式，而甲板对称方向则带有唯一的典范权重1/2形式。缺陷不变量满足||Xi||²<v₊,v₊>=144，且等于L(f,2)/(48π²<f,f>)，精确到31位。在截面层，Petersson几何是刚性的：S_{3/2}的Gram矩阵是精确有理形式的单个超越数倍，该超越数被识别为3Γ(1/3)³/(2^{7/3}π²)，精确到40位。权重3/2的阴影范数属于Chowla-Selberg环，权重5/2的范数则被排除在外。
英文摘要
The four orders of the quaternion algebra B6 carry four reflective wall data on lattices of signature (3,2); three integrate to Borcherds denominators and one is obstructed. The failed denominator survives as a weakly harmonic Maass form, and we prove its shadow is a Hecke eigenform on the line of the newform 6.4.a.a, with zero twist component. The mechanism is invariance selection: the obstruction functional is invariant under the discriminant isometry group, whose invariants in S_{5/2} are one-dimensional; the same mechanism, verified at quaternion discriminants 10 and 22, places the shadows there on 10.4.a.a and 22.4.a.c. On the weight-1/2 layer we prove a determination theorem: the canonical form exists and is unique precisely when the obstruction space vanishes, and among the 71 discriminants below 230 this happens exactly for D in {6, 10, 22}, the genus-zero compact Shimura curves, whose maximal-order ternaries are reflective with integral Weyl chambers of ranks 3, 4, 4; completeness beyond that range is reduced to an estimate on a quadratic Dedekind-type sum, given a bound on the Gauss-sum term. Parity confines this layer to odd channels; on the obstructed orientation the section layer is obstructed outside an explicit 40-element locus of orientations (a double shadow: CM, 36.2.a.a, at weight 3/2 and newform at weight 5/2), while the deck-symmetric directions instead carry a unique canonical weight-1/2 form. The defect invariant satisfies ||Xi||^2 <v_+,v_+> = 144 exactly and equals L(f,2)/48 pi^2 <f,f> to 31 digits. On the section layer the Petersson geometry is rigid: the Gram matrix of S_{3/2}(rho_4) is a single transcendental multiple of an exact rational form, and that transcendental is identified, to 40 digits, as 3 Gamma(1/3)^3 / (2^{7/3} pi^2): the weight-3/2 shadow norms lie in the Chowla-Selberg ring; the weight-5/2 norm is numerically excluded from it.
