The following problem is listed here: <http://www-personal.umich.edu/~erman/Papers/Questions2.pdf> and attributed to Vistoli:


Let $\mathcal A\_g$ denote the moduli stack of principally polarized abelian varieties over a field $k$, let $A\_g$ denote the associated coarse moduli space, and $K(A\_g)$ its function field. What is the minimal degree of a **finite** extension $L/K(A\_g)$ so that there exists a commutative diagram:
$$\begin{matrix}\operatorname{Spec}L&\to&\mathcal A\_g\cr\downarrow&&\downarrow&&?\cr\operatorname{Spec}K(A\_g)&\to&A\_g\end{matrix}$$
The paper ["Essential dimension of moduli of curves and other algebraic stacks"](http://arxiv.org/abs/0907.0924) by Brosnan--Reichstein--Vistoli--Fakhruddin is listed as a reference. The last section of this paper (by Fakhruddin) calculates the essential dimension of the stack $\mathcal A\_g$. My question is: how is the notion of essential dimension of a stack related to the problem posed by Vistoli?


For reference, the **essential dimension** of a functor $F:\mathfrak F\mathfrak i\mathfrak e\mathfrak l\mathfrak d\mathfrak s\_{/k}\to\mathfrak S\mathfrak e\mathfrak t\mathfrak s$ is the minimal integer $n$ such that every element of $F(K/k)$ comes from an element of $F(L/k)$ for some $L/k$ of **transcendence degree** at most $n$ over $k$ (if no such $n$ exists, then the essential dimension of $F$ is infinite). The essential dimension of a functor $F:\mathfrak F\mathfrak i\mathfrak e\mathfrak l\mathfrak d\mathfrak s\_{/k}\to\mathfrak G\mathfrak r\mathfrak o\mathfrak u\mathfrak p\mathfrak o\mathfrak i\mathfrak d\mathfrak s$ is just the essential dimension of $\left|F\right|$ (where $\left|\cdot\right|:\mathfrak G\mathfrak r\mathfrak o\mathfrak u\mathfrak p\mathfrak o\mathfrak i\mathfrak d\mathfrak s\to\mathfrak S\mathfrak e\mathfrak t\mathfrak s$ denotes taking isomorphism classes).



Essential dimension and the moduli space of abelian varieties
The two notions are related using Theorems 4.1 and 6.1 of the paper of Brosnan, Reichstein and Vistoli: 


Theorem 6.1 reduces the computation of the essential dimension of the stack to that of the generic gerbe $\mathcal{X}\_g$. Theorem 4.1 says that the essential dimension of $\mathcal{X}\_g$ is $\mathrm{cd}(\mathcal{Y}\_g) - 1$, where $\mathrm{cd}(\mathcal{Y}\_g)$ is the canonical dimension of the Brauer--Severi variety $\mathcal{Y}\_g$ associated to $\mathcal{X}\_g$. Since the index of $\mathcal{Y}\_g$ is a power of $2$ (as the order is $2$), the last line of Theorem 4.1 implies that $\mathrm{cd}(\mathcal{Y}\_g) + 1$ is equal to the index, which is by definition the degree of the field $L$.


