  A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$
automorphism which acts symplectically on the K3 surface and acts by shifting
by an $N$-torsion point on the elliptic curve $E$. We conjecture that the
primitive Donaldson-Thomas partition function of elliptic CHL models is a
Siegel modular form, namely the Borcherds lift of the corresponding
twisted-twined elliptic genera which appear in Mathieu moonshine. The
conjecture matches predictions of string theory by David, Jatkar and Sen. We
use the topological vertex to prove several base cases of the conjecture. Via a
degeneration to $\mathrm{K3} \times \mathbb{P}^1$ we also express the DT
partition functions as a twisted trace of an operator on Fock space. This
yields further computational evidence. An extension of the conjecture to
non-geometric CHL models is discussed.
  We consider CHL models of order $N=2$ in detail. We conjecture a formula for
the Donaldson-Thomas invariants of all order two CHL models in all curve
classes. The conjecture is formulated in terms of two Siegel modular forms. One
of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet
appeared in physics. This discrepancy is discussed in an appendix with Sheldon
Katz.
