1
$\begingroup$

References are to K. Kitagawa, "On standard $p$-adic $L$-functions of families of elliptic cusp forms", Contemp. Math. 165, 1994.

Let $\mathcal O$ be the ring of integers in a finite extension of $\mathbb Q_p$ and $N$ an integer prime to $p$. For $n\ge0$, denote by $\overline{\mathcal{MS}}^{\mathrm{ord}}(n,\mathcal O)$ the ordinary part of the $p$-adic completion of the direct limit of the spaces of (classical) modular symbols of weight $n+2$ and level $Np^r$ for $r\to\infty$, as defined by Kitagawa (§5.1). This space carries an action of the group $\Gamma=1+p\mathbb Z_p$.

Kitagawa proves (Thm. 5.3) that there is a canonical isomorphism $$ \overline{\mathcal{MS}}^{\mathrm{ord}}(n,\mathcal O) \cong \overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathcal O)(\kappa^n) $$ with $\kappa\colon\Gamma\rightarrow\mathcal O^\times$ the tautological inclusion and $(\kappa^n)$ meaning that the action of $\Gamma$ has been twisted by $\kappa^n$. Then he claimes (implicitly, in the proof of Thm. 5.6) that $$ \overline{\mathcal{MS}}^{\mathrm{ord}}(n,\mathcal O)[\varepsilon] \cong \overline{\mathcal{MS}}^{\mathrm{ord}}(0,\mathcal O)[\varepsilon\kappa^n] $$ for any finite order character $\varepsilon$ of $\Gamma$, where $[\cdot]$ denotes a character eigenspace.

But if we have an $\mathcal O[\![\Gamma]\!]$-module $M$ and characters $\chi$ and $\varepsilon$, then an easy calculation shows that $M(\chi)[\varepsilon]=M[\chi^{-1}\varepsilon]$, which seems to contradict the above. Since the above statement is essential for the control theory of $p$-adic modular symbols and thus for Kitagawa's construction, it is hopefully true. What is wrong here?

See also the related question Control theory for Kitagawa's $\Lambda$-adic modular symbols.

$\endgroup$

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.