Reductions of Galois representations and the theta operator

Eknath Ghate, Arvind Kumar

Published 2019-06-25Version 1

Let $p\ge 5$ be a prime, and let $f$ be an eigenform of weight at least $2$ and level coprime to $p$ of finite slope $\alpha$. Let $\bar{\rho}_f$ denote the mod $p$ Galois representation associated with $f$ and $\omega$ the mod $p$ cyclotomic character. Under some mild assumptions, we prove that there exists an eigenform $g$ of weight at least $2$ and level coprime to $p$ of slope $\alpha+1$ such that $$\bar{\rho}_f \otimes \omega \simeq \bar{\rho}_g,$$ up to semisimplification. The proof uses Hida-Coleman families and the theta operator acting on overconvergent forms. The structure of the reductions of the local Galois representations associated to cuspforms with slopes in the interval $[0,1)$ were determined by Deligne, Buzzard, Gee and for slopes in $[1,2)$ by Bhattacharya, Ghate, Rai and Rozensztajn. We show that these reductions, in spite of their somewhat complicated behavior, are compatible with the displayed equation above. Moreover, the displayed equation above allows us to predict the shape of the reductions of a class of Galois representations attached to eigenforms of slope larger than $2$. Finally, recall that Wan has given upper bounds on the radii of Coleman families. The methods of this paper allow us to obtain lower bounds on the radii of certain Coleman families.