Let p be an odd prime and k a non-negative integer. Let N be a positive integer such that $$p \not \mid N$$ and $$\lambda $$ a Dirichlet character modulo N. We obtain quantitative lower bounds for the number of Dirichlet character $$\chi $$ modulo F with the critical Dirichlet L-value $$L(-k,\lambda \chi )$$ being p-indivisible. Here $$F \rightarrow \infty $$ with $$(N,F)=1$$ and $$p\not \mid F\phi (F)$$ . We explore the indivisibility via an algebraic and a homological approach. The latter leads to a bound of the form $$F^{1/2}$$ . The p-indivisibility yields results on the distribution of the associated p-Selmer ranks. We also consider an Iwasawa variant. It leads to an explicit upper bound on the lowest conductor of the characters factoring through the Iwasawa $${{\mathbb {Z}}}_{\ell }$$ -extension of $${{\mathbb {Q}}}$$ for an odd prime $$\ell \ne p$$ with the corresponding critical L-value twists being p-indivisible.

中文翻译：

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Dirichlet L 值模 p 的定量非消失

令 p 为奇素数且 ka 为非负整数。设 N 是一个正整数，使得 $$p \not \mid N$$ 和 $$\lambda $$ 是一个 Dirichlet 字符模 N。我们得到 Dirichlet 字符 $$\chi $$ 模 F 的数量的定量下界临界 Dirichlet L 值 $$L(-k,\lambda \chi )$$ 是 p 不可分的。这里 $$F \rightarrow \infty $$ 与 $$(N,F)=1$$ 和 $$p\not \mid F\phi (F)$$ 。我们通过代数和同调方法探索不可分割性。后者导致 $$F^{1/2}$$ 形式的边界。p-不可分割性产生了相关 p-Selmer 等级分布的结果。我们还考虑了 Iwasawa 变体。