A Conjecture of Coleman on the Eisenstein Family

We prove for primes $p\ge 5$ a conjecture of Coleman on the analytic continuation of the family of modular functions $\frac{E_{\kappa }^{\ast }}{V(E_{\kappa }^{\ast })}$ derived from the family of Eisenstein series $E_{\kappa }^{\ast }$. 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.