Fix a level $N \geq 3$ and denote by $\Gamma(N)\subset SL(2,\mathbb{Z})$ the subgroup of matrices which are congruent to the identity modulo $N$. The open modular curve $Y(N)$ corresponding to $\Gamma(N)$ classifies elliptic curves with full level $N$-sructure. There is then a universal elliptic curve $\pi:E\to Y(N)$. Everything is defined over the extension $L$ of $\mathbb{Q}_p$ generated by the $N$th roots of unity (Fix a prime $p>2$, $(p,N)=1$).
View the structure $\mathcal{O}$ as a crystalline sheaf on $E$. Set $\mathbb{V}=R^1\pi_{*,crys}(\mathcal{O})$. Then $M:=H^1_{crys}(Y(N), \operatorname{Sym}^2(\mathbb{V}))$ is a $F$-crystal, has weight $0$ and $3$. it has filtraiton $F^0\supsetneq F^1=F^2=F^3\supsetneq 0$. Let $\phi$ denote the frobenius on $M$ then it induce a Hasse-witt mod $p$ map $M/(pM+F^1)\to M/(pM+F^1)$. My question is:
Is this map bijective? Is there any reference? Or maybe for some primes $p$ it is bijective, for other primes it is not ?
Thank you!
