Question

In the paper Construction of some families of 2-dimensional crystalline representations by Berger-Li-Zhu, a very explicit description is given for the $(\phi,\Gamma)$-module attached to some 2-dimensional crystalline representations. In particular, if one takes the (local) Galois representation arising from a modular form $f$ of level prime to $p$ with $a_p(f)=0$, then a very simple formula is given for the $\phi$ and $\Gamma$ action on the $(\phi,\Gamma)$-module $D(V)$.

Further, if one takes the $\psi=1$ invariants of $D(V)$ one should recover $H^1_{Iw}(V)$. I was hoping to see these $\psi=1$ invariants completely explicitly in the $a_p(f)=0$ case. In trying to do so, it appears to boil down to having to find a function $P$ in some ring like $B_{rig}$ satisfying $\psi^2(P) = -P$.

So here's my question: is there any way to find an explicit $P$ satisfying this equation?

Answer 1

It is not so easy to construct an explicit basis for the module of $\psi$-invariants, sadly, although we know that is a $\Lambda(\Gamma)$-module of rank $\dim V$. What is much easier is to compose with the map $1 - \varphi$. The module $C(V) = (1 - \varphi) \mathbb{D}(V)^{\psi = 1}$, which Colmez calls the "coeur" (heart) of $V$, is contained in $\mathbb{D}(V)^{\psi = 0}$ and is isomorphic to $H^1_{\mathrm{Iw}}(V) / V^{H_{\mathbb{Q}_p}}$. When $V$ is crystalline and the Hodge--Tate weights are $\ge 0$, $C(V)$ actually lands in $$(\mathbb{B}^+_{\mathrm{rig}, \mathbb{Q}_p})^{\psi = 0} \otimes \mathbb{D}_{\mathrm{cris}}(V) \cong \mathcal{H}(\Gamma) \otimes \mathbb{D}_{\mathrm{cris}}(V),$$ where $\mathcal{H}(\Gamma)$ is the ring of distributions on $\Gamma$.

In my paper with Sarah Zerbes and Antonio Lei (in Asian J Math), we showed how to find a basis for $C(V)$ using a basis for the Wach module of $V$. In particular, for modular forms with $a_p = 0$ one can write down explicit generators of $C(V)$ using the half-logarithm distributions of Pollack. For $a_p \ne 0$, but when the valuation of $a_p$ is large enough that the Berger--Li--Zhu description of the Wach module applies, then their basis of the Wach module again gives a (slightly less explicit) description of $C(V)$.