Computing Crystalline Deformation Rings via the Taylor-Wiles-Kisin Patching Method

Fix a prime number $p>2.$ Let $\bar{r} : Gal(\overline{Q}_p/Q_p)\to GL_2(\overline{F}_p)$ be an absolutely irreducible residual representation. In this paper, we describe an algorithm to compute arbitrarily close approximations to the non-framed fixed-determinant crystalline deformation ring $R^k_\bar{r}$ of $\bar{r}$ whose $\overline{Q}_p$-points parametrize crystalline representations with Hodge--Tate weights $(0,k-1)$ using the Taylor--Wiles--Kisin patching. We give an implementation of this algorithm in Magma and Python (with Sagemath imported). Based on the data we have collected, we formulate a conjecture on the Hilbert series of the special fiber of $R^k_\bar{r}$ when $k=2+n(p-1)$ for some non-negative integer $n$. The conjectural formula implies that the Hilbert series goes to $(1-x)^{-3}$ as $n$ tends to $\infty.$ This aligns with the expectation that, as $n$ grows, the special fiber of $R^k_\bar{r}$ gradually ``fills out" that of the universal fixed-determinant deformation ring of $\bar{r},$ which is a formal power series ring in three variables. We also formulate a conjecture on when $R^k_\bar{r}$ is Gorenstein.