Commensurability classes of fake quadrics

Abstract

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop a number of new techniques that explicitly bound the arithmetic invariants of a fake quadric and more generally of an arithmetic manifold of bounded volume arising from a form of ${{\,\mathrm{SL}\,}}_2$ over a number field.

Appendix

In this appendix we list maximal arithmetic subgroups of ${{\,\mathrm{PGL}\,}}_2({\mathbb {R}}) \times {{\,\mathrm{PGL}\,}}_2({\mathbb {R}})$ that may contain the stable subgroup of the fundamental group of an irreducible fake quadric. Since the number of these commensurability classes is large, we employ some shortcuts in how we record our results, which we now describe (Tables 1, 2, 3, 4, 5).

We group maximal stable arithmetic subgroups first by the data

$$\begin{aligned} n, d, D, N \end{aligned}$$

where $n=[k:{\mathbb {Q}}]$ is the degree of the underlying totally real field k and d is its discriminant (for the commensurability classes below, the discriminant uniquely determines the field); the associated quaternion algebra B is specified by the absolute norm $D={{\,\mathrm{N}\,}}({\mathfrak {D}})$ of its discriminant ${\mathfrak {D}}$, and the Eichler order ${\mathcal {E}}\subset B$ is specified by the absolute norm $N={{\,\mathrm{N}\,}}({\mathfrak {N}})$ of its level ${\mathfrak {N}}$. There are only finitely many possibilities for $k,B,{\mathcal {O}}$ with this data, and they are explicitly given in the computer readable output available online [26].

In each line of the tables, we provide a bit more data about the groups (which in some cases depends on more than just the data above, so the data n, d, D, N may be repeated). First, we compute the covolume of the maximal stable arithmetic group with the specified data. Second, we compute the index of the maximal holomorphic stable group inside the maximal stable group—in nearly all cases, this index is 4 (coming from elements acting by orientation-reversing isometry on each of the two factors of ${\mathcal {H}}$). Third, we compute a divisor of the least common multiple of the orders of the elements of finite order in $\Gamma _{S,{\mathcal {O}}}$. Fourth, we compute the number $0 \le \nu \le |S|$ that appears in (5.14) using class field theory [29, p. 356]. Finally, the last column records $\star$ if it is guaranteed that $\Gamma = \Gamma _{st}$; otherwise, we leave this entry blank, indicating that it is possible that there are unstable fake quadric groups $\Gamma$ with this data.

Remark 5.18

We now make some remarks on the discrepancies between our work and D$\check{\hbox {z}}$ambić’s [18] on fake quadrics defined over quadratic fields. Any lattices in our paper that are not stable will not appear in [18]. Unfortunately, our tables also differ for stable lattices. We found that the entry $[{\mathbb {Q}}(\sqrt{5}), v_2 v_{31}, \emptyset , 2]$ in [18, Thm. 3.15], and similarly for $v_{31}^\prime$, cannot produce fake quadrics, even in the generality discussed in this paper.

We quickly explain how our index calculations should differ from those of [18] when ${\mathfrak {N}}= 1$, i.e., for $\Gamma _{\mathcal {O}}$. It is easy to see, using Eichler’s theorem on norms, that the index of $\Gamma _{\mathcal {O}}^+$ in $\Gamma _{\mathcal {O}}$ is equal to 2 when the narrow class number $h^+$ equals the class number h, since we can find an element of ${\mathcal {O}}^*$ with reduced norm to k that is negative at each real place of k, but cannot find one that is negative at exactly one real place. Otherwise, $h^+ = 2 h$ and $\Gamma _{\mathcal {O}}^+$ has index 4 in $\Gamma _{\mathcal {O}}$, since we can find elements of ${\mathcal {O}}^*$ with reduced norm of chosen sign at each real embedding. When there is the possibility that the lattice has proper stable subgroup, we report the index in $\Gamma _{\mathcal {O}}$ of a (potential) lattice of covolume $32 \pi ^2$. In this case, our index should be 4 times the index I from [18] when $h^+ = h$ and 8 times D$\check{\hbox {z}}$ambić’s I when $h^+ = 2 h$. When $\Gamma _{\mathcal {O}}$ is stable, we report the index of a (potential) subgroup of covolume $16 \pi ^2$, so our index should be 2I when $h^+ = h$ and 4I when $h^+ = 2 h$. This, plus the index of $\Gamma _{\mathcal {O}}^+$ in $N\Gamma _{\mathcal {O}}^+$ (in the notation from [18]), should account for the difference between our index and Dzambic’s.

Table 1 Commensurability classes for degree $n=2$

Table 2 Commensurability classes for degree $n=3$

Table 3 Commensurability classes for degree $n=4$

Table 4 Commensurability classes for degree $n=5$

Table 5 Commensurability classes for degree $n=6$