  Recent studies have proposed that most warm Jupiters (WJs, giant planets with
semi-major axes in the range of 0.1-1 AU) probably form in-situ, or arrive in
their observed orbits through disk migration. However, both in-situ formation
and disk migration, in their simplest flavors, predict WJs to be in
low-eccentricity orbits, in contradiction with many observed WJs that are
moderately eccentric (e=0.2-0.7). This paper examines the possibility that the
WJ eccentricities are raised by secular interactions with exterior giant planet
companions, following in-situ formation or migration on a circular orbit.
Eccentricity growth may arise from an inclined companion (through Lidov-Kozai
cycles), or from an eccentric, nearly coplanar companion (through apsidal
precession resonances). We quantify the necessary conditions (in terms of the
eccentricity, semi-major axis and inclination) for external perturbers of
various masses to raise the WJ eccentricity. We also consider the sample of
eccentric WJs with detected outer companions, and for each system, identify the
range of mutual inclinations needed to generate the observed eccentricity. For
most systems, we find that relatively high inclinations (at least $\sim
40^\circ$) are needed so that Lidov-Kozai cycles are induced; the observed
outer companions are typically not sufficiently eccentric to generate the
observed WJ eccentricity in a low-inclination configuration. The results of
this paper place constraints on possibly unseen external companions to
eccentric WJs. Observations that probe mutual inclinations of giant planet
systems will help clarify the origin of eccentric WJs and the role of external
companions.
