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Let $F$ be a totally real field of even degree and let $f$ be a Hilbert cusp form of level $\mathfrak{n}$ that is an eigenform for Hecke operators. Let $p$ be an odd prime number and $\mathfrak{p}$ a prime of $F$ above $p$. Let $\rho$ be a continuous representation associated to $f$ at $\mathfrak{p}$ (that satisfies properties on det and trace). Let $S$ be the set of primes of $F$ dividing $p\mathfrak{n}$ and $X_S$ the Galois group of the maximal abelian pro-$p$-extension of $F$ that is unramified outside $S$.

My question is whether possible to obtain information about the structure of the $\mathbb{Z}_p$-torsion of $X_S$ from the study of the image of $\rho$?

Thank you in advance for any comments.

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  • $\begingroup$ To be clear, is $\rho$ and $\ell$-adic representation for $\ell \ne p$? $\endgroup$ Commented May 26 at 1:59
  • $\begingroup$ @Kimball it is a $\mathfrak{p}$-adic representation, if I understand you question well. $\endgroup$ Commented May 26 at 22:19

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