# Essential dimension and the moduli space of abelian varieties

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" 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).

• The two notions are related using Theorems 4.1 and 6.1 of the paper: Theorem 6.1 reduces the computation of the essential dimension of the stack to that of the generic gerbe. Theorem 4.1 gives a formula for the essential dimension of the generic gerbe in terms of the canonical dimension of the associated Brauer--Severi variety. In this particular case (see the last line of Theorem 4.1) the canonical dimension is given in terms of the index of the Brauer--Severi variety, and this is the degree of the field $L$. – ulrich Feb 25 '16 at 5:53

## 1 Answer

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 $L$.