Oeljeklaus–Toma manifolds and arithmetic invariants

Abstract

We consider Oeljeklaus–Toma manifolds coming from number fields with precisely one complex place. Our general theme is to relate the geometry to the arithmetic. We show that just knowing the fundamental group allows us to recover the number field. We also show that this fails if there are more complex places. The first homology turns out to be related to an interesting ideal. We compute the volume in terms of the discriminant and regulator of the number field. Is there a conceptual reason for this? We explore this and see what happens if we (entirely experimentally) regard them as “baby siblings” of hyperbolic manifolds coming from number fields.

Appendix

Computer Code 1

The computations underlying Example 2 can be confirmed in an automated fashion by computer algebra systems. The following code is written for SAGE [17], largely using PARI/GP [19]. Firstly, we confirm that S was a generator of the group of units (up to torsion):

This computation can also be done by hand using the Minkowski bounds; but remember that this verification was actually not needed for the validity of the example. Next, we check the crucial fact that the order $J_{i}$ is maximal, i.e. that it is the ring of integers:

Adapt the minimal polynomial for t to check both cases $i=2,3$. Finally, we check that the compositum of the Galois closures has degree 216:

Of course it would not be particularly hard to perform this computation by hand, just a bit tedious.

Computer Code 2

We discuss the determination of the ideal J(U), Definition 1, by computer. We have used this for our Example 1. The following code runs through the number fields generated by the minimal polynomials $Z^{3}-Z+h$, whenever these are irreducible, for $h=1,\ldots ,9$. In this particular case these number fields have $s=t=1$ real resp. complex places, so $\mathcal {O}_{K}^{\times } \simeq \left\langle -1\right\rangle \times \mathbf {Z}\left\langle u\right\rangle$, where u is a fundamental unit. For these minimal polynomials the single real embedding of the fundamental unit always happens to be negative. This follows from Descartes’ Rule of Signs: The polynomial rewritten in $-Z$ is $-Z^{3}+Z+h$, which has precisely one sign change among its coefficients. Therefore, it must have a single negative real root. Hence, $\mathcal {O} _{K}^{\times ,+}\simeq \mathbf {Z}\left\langle -u\right\rangle$ and the ideal $J(\mathcal {O}_{K}^{\times ,+})$ is generated by the single elemet $1-(-u)=1+u$ by Lemma 2.

Note that SAGE always returns the unit group in the format so that U.gen(0) is the torsion generator and U.gen(1) the non-torsion generator. Hence, in this particular case the ideal J needs to be generated by 1+U.gen(1). This code can easily be adapted to similar computations. For example, for the polynomials $Z^{7}-Z-h$ we will have $s=1$ and $t=3$ real resp. complex places. Any such polynomial has exactly one sign change in its coefficients, so by Descartes’ Rule it has precisely one positive real root. Hence, $\mathcal {O}_{K}^{\times } =\{\pm 1\}\times \mathcal {O}_{K}^{\times ,+}$ and therefore $J(\mathcal {O} _{K}^{\times ,+})$ is generated by the elements $1-u$, where u runs through the generators of $\mathcal {O}_{K}^{\times ,+}$, again by Lemma 2. Replace the definition of T by

T = [(1-U.gen(i)) for i in range(1,len(U.gens()))],

since we now will have several generators. We discard the generator at i=0 since this is again the torsion generator.