of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ such that $\epsilon\delta^{-1}$ is an algebraic character of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$ and such that $\epsilon$ is strongly linked to $\delta$ in the sense of [@HumBGG §5.1] (as modules over the Lie algebra of $T(F^+\otimes_{{\mathbb{Q}}}{\mathbb{Q}}_p)$). Then $((\rho_v)_{v\in S_p},\delta)$ is also a point on $X_p({\overline{\rho}}_x)$.

We also prove that the equality (\[mainequality\]) for all $v\in S_p$ (and thus the modularity lifting conjectures) imply that the initial Hecke eigenvariety is itself [*singular*]{} at $x$ as soon as the Weyl element $w_{x,v}$ is [*not*]{} a product of distinct simple reflections for some $v\in S_p$, see Corollary \[singhecke\].

Let us now outline the strategy of the proofs of Theorems \[mainintro2\] and \[mainlocal\].

The proof of Theorem \[mainlocal\] crucially uses results of Bergdall ([@Bergdall]), together with a fine analysis of the various conditions on the infinitesimal deformations of $\rho_{x,v}$ carried by vectors in $T_{X,x_v}$, see §\[wedgesection\] and §\[endofproof\]. Very recently, Bergdall proved an analogous bound for the dimension of the tangent space of the initial Hecke eigenvariety at $x$ assuming standard vanishing conjectures on certain Selmer groups ([@Bergdraft]).

The proof of Theorem \[mainintro2\] makes use of the patched eigenvariety $X_p({\overline{\rho}}_x)$ constructed in [@BHS] by applying Emerton’s construction of eigenvarieties [@Emerton] to the locally analytic vectors of 