show/hide this revision's text 3 I made a grievous error confusing vertices and edges. Whoops!

Inkspot is indeed correct that the component graphs are indeed not generally trees.

As you seem to have deduced for yourself, Cerednik-Drinfeld uniformization is a highly nontrivial concept, and it really helps to have some examples to set it in your mind. Most helpful in this direction is Ogg's "Mauvaise réduction des courbes de Shimura" where he draws out a few of these dual graphs.

In general the dual graphs have $2h$ edges vertices $x$ where $h$ is the class number of $\mathcal{O}_x$, a level $H$ Eichler order in $D$ (your totally definite quaternion algebra, so note there's a choice of which $x$ to make here)here, but as long as the level is squarefree all orders are hereditary and it doesn't make a difference).

An orbit-stabilizer theorem computation then shows that for instance when $B$ is a quaternion algebra over $\mathbf{Q}$, $p+1$ (the size of the set of edges $y$ stemming from a particular vertex $x$ in the Bruhat-Tits tree) is equal to $\sum_{e(y) = x} \frac{\mathcal{O}_x^\times}{\mathcal{O}_y^\times}$. So it's not just that there are edges, but we know exactly how many there are! (For a readable account of details of this, see Kurihara's paper on Equations defining Shimura Curves)

Also, if I may take issue with 1.(ii) and 1.(iii), you've given a good description of the special fiber over $\overline \kappa_v$ (which is what I'm taking the dual graph to represent the data for), not necessarily $\kappa_v$. What you've claimed is that the special fiber is a Mumford Curve, that is, the transverse union of a number of copies of $\mathbb{P}^1$'s. The truth is that the special fiber is a quadratic twist of a Mumford curve, where the Galois action is not simply given by the $| \kappa_v|$-Frobenius, but where the action of Frobenius is identified with the action of the Atkin-Lehner operator $w_p$, which interchanges some of the components (if you want to think about the graph, its edge vertex set can be partitioned into $ {x_1, \dots , x_h, x_1', \dots, x_h'}$ where $w_p(x_i) = x_i'$).

All that said, some of the best advice I've heard for trying to understand this stuff is to first completely understand what happens when $v$ divides the LEVEL because in that case the moduli problem is much easier (if an abelian variety here is isogenous to a product of supersingular elliptic curves, it's isomorphic to a product of supersingular elliptic curves)

Here are a few additional references:

http://www.math.mcgill.ca/cfranc/documents/bctranslation.pdf (a translation of Boutot-Carayol)

http://math.berkeley.edu/~ribet/Articles/bimodules.pdf (this includes a somewhat more intuitive description of the components of the Mumford curve)

http://math.uga.edu/~pete/thesis.pdf (a comprehensive introduction to Shimura Curves and the action of the Atkin-Lehner group)

http://www.springerlink.com/content/gj8365486214l141/ (this is Oort's "which abelian surfaces are products of supersingular elliptic curves", see also his book on moduli of supersingular abelian varieties, as when $v$ ramifies in $B$, you're asking a question about moduli of supersingular abelian varieties, see the appendix on Honda-Tate theory to the thesis above)
show/hide this revision's text 2 deleted 16 characters in body

Inkspot is indeed correct that the component graphs are indeed not generally trees.

As you seem to have deduced for yourself, Cerednik-Drinfeld uniformization is a highly nontrivial concept, and it really helps to have some examples to set it in your mind. Most helpful in this direction is Ogg's "Mauvaise réduction des courbes de Shimura" where he draws out a few of these dual graphs.

In general the dual graphs have $2h$ edges where $h$ is the class number of a level $H$ Eichler order in $D$ (your totally definite quaternion algebra, so note there's a choice to make here).

Also, if I may take issue with 1.(ii) and 1.(iii), you've given a good description of the special fiber over $\overline \kappa_v$ (which is what I'm taking the dual graph to represent the data for), not necessarily $\kappa_v$. What you've claimed is that the special fiber is a Mumford Curve, that is, the transverse union of a number of copies of $\mathbb{P}^1$'s. The truth is that the special fiber is a quadratic twist of a Mumford curve, where the Galois action is not simply given by the $| \kappa_v|$-Frobenius, but where the action of Frobenius is identified with the action of the Atkin-Lehner operator $w_p$, which interchanges some of the components (if you want to think about the graph, its edge set can be partitioned into $ {x_1, \dots , x_h, x_1', \dots, x_h'}$ where $w_p(x_i) = x_i'$).

All that said, some of the best advice I've heard for trying to understand this stuff is to first completely understand what happens when $v$ divides the LEVEL because in that case the moduli problem is much easier (if an abelian variety here is isogenous to a product of supersingular elliptic curves, it's isomorphic to a product of supersingular elliptic curves)

Here are a few additional references:

http://www.math.mcgill.ca/cfranc/documents/bctranslation.pdf (a translation of Boutot-Carayol)

http://math.berkeley.edu/~ribet/Articles/bimodules.pdf (this includes a nonstandard but somewhat more intuitive description of the components of the Mumford curve)

http://math.uga.edu/~pete/thesis.pdf (a comprehensive introduction to Shimura Curves and the action of the Atkin-Lehner group)

http://www.springerlink.com/content/gj8365486214l141/ (this is Oort's "which abelian surfaces are products of supersingular elliptic curves", see also his book on moduli of supersingular abelian varieties, as when $v$ ramifies in $B$, you're asking a question about moduli of supersingular abelian varieties, see the appendix on Honda-Tate theory to the thesis above)
show/hide this revision's text 1

Inkspot is indeed correct that the component graphs are indeed not generally trees.

As you seem to have deduced for yourself, Cerednik-Drinfeld uniformization is a highly nontrivial concept, and it really helps to have some examples to set it in your mind. Most helpful in this direction is Ogg's "Mauvaise réduction des courbes de Shimura" where he draws out a few of these dual graphs.

In general the dual graphs have $2h$ edges where $h$ is the class number of a level $H$ Eichler order in $D$ (your totally definite quaternion algebra, so note there's a choice to make here).

Also, if I may take issue with 1.(ii) and 1.(iii), you've given a good description of the special fiber over $\overline \kappa_v$ (which is what I'm taking the dual graph to represent the data for), not necessarily $\kappa_v$. What you've claimed is that the special fiber is a Mumford Curve, that is, the transverse union of a number of copies of $\mathbb{P}^1$'s. The truth is that the special fiber is a quadratic twist of a Mumford curve, where the Galois action is not simply given by the $| \kappa_v|$-Frobenius, but where the action of Frobenius is identified with the action of the Atkin-Lehner operator $w_p$, which interchanges some of the components (if you want to think about the graph, its edge set can be partitioned into $ {x_1, \dots , x_h, x_1', \dots, x_h'}$ where $w_p(x_i) = x_i'$).

All that said, some of the best advice I've heard for trying to understand this stuff is to first completely understand what happens when $v$ divides the LEVEL because in that case the moduli problem is much easier (if an abelian variety here is isogenous to a product of supersingular elliptic curves, it's isomorphic to a product of supersingular elliptic curves)

Here are a few additional references:

http://www.math.mcgill.ca/cfranc/documents/bctranslation.pdf (a translation of Boutot-Carayol)

http://math.berkeley.edu/~ribet/Articles/bimodules.pdf (this includes a nonstandard but somewhat more intuitive description of the components of the Mumford curve)

http://math.uga.edu/~pete/thesis.pdf (a comprehensive introduction to Shimura Curves and the action of the Atkin-Lehner group)

http://www.springerlink.com/content/gj8365486214l141/ (this is Oort's "which abelian surfaces are products of supersingular elliptic curves", see also his book on moduli of supersingular abelian varieties, as when $v$ ramifies in $B$, you're asking a question about moduli of supersingular abelian varieties, see the appendix on Honda-Tate theory to the thesis above)