论文标题
科尔曼在爱森斯坦家族上的猜想
A Conjecture of Coleman on the Eisenstein Family
论文作者
论文摘要
我们证明了Primes $ p \ ge 5 $ Coleman对模块化函数家族的分析延续的猜想$ \ frac {e^\ast_κκ} {猜想的确切，定量公式涉及一定的$ p $，具体取决于常数。我们以一个示例表明，猜想与科尔曼猜想的常数一般不能对所有素数保持。另一方面，在所有情况下，我们给出的常数也不是最佳的。该猜想是由于Buzzard和Kilford与Roe的某些中央声明的联系而动机，以及ROE，关于Primes $ 2 $和3美元的“ Halo”猜想。我们展示了我们的结果如何概括这些陈述并评论可能的未来发展。
We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{E^\ast_κ}{V(E^\ast_κ)}$ derived from the family of Eisenstein series $E^\ast_κ$. The precise, quantitative formulation of the conjecture involved a certain on $p$ depending constant. We show by an example that the conjecture with the constant that Coleman conjectured cannot hold in general for all primes. On the other hand, the constant that we give is also shown not to be optimal in all cases. The conjecture is motivated by its connection to certain central statements in works by Buzzard and Kilford, and by Roe, concerning the "halo" conjecture for the primes $2$ and $3$, respectively. We show how our results generalize those statements and comment on possible future developments.
