Values of quaternionic modular forms are related to twisted central $L$-values via periods and a theorem of Waldspurger. In particular, certain twisted $L$-values must be non-vanishing for forms with no zeroes. Here we study, theoretically and computationally, zeroes of definite quaternionic modular forms of trivial weight. Local sign conditions force certain forms to have trivial zeroes, but we conjecture that almost all forms have no nontrivial zeroes. In particular, almost all forms with appropriate local signs should have no zeroes. We show these conjectures follow from a conjecture on the average number of Galois orbits, and give applications to (non)vanishing of $L$-values.

中文翻译：

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四元模形式的零点和中心 L 值

四元模形式的值通过周期和 Waldspurger 定理与扭曲的中心 $L$ 值相关。特别是，对于没有零的表单，某些扭曲的 $L$ 值必须不消失。在这里，我们从理论上和计算上研究了平凡权重的确定四元模形式的零点。局部符号条件迫使某些形式具有平凡零，但我们推测几乎所有形式都没有非平凡零。特别是，几乎所有具有适当局部符号的形式都不应有零。我们展示了这些猜想来自对伽罗瓦轨道平均数的猜想，并应用到（非）消失的 $L$ 值。