  The 4D compact U(1) gauge theory has a well-established phase transition
between a confining and a Coulomb phase. In this paper, we revisit this model
using state-of-the-art Monte Carlo simulations on anisotropic lattices. We map
out the coupling-temperature phase diagram, and determine the location of the
tricritical point, $T/K_0 \simeq 0.19$, below which the first-order transition
is observed. We find the critical exponents of the high-temperature
second-order transition to be compatible with those of the 3-dimensional $O(2)$
model. Our results at higher temperatures can be compared with literature
results and are consistent with them. Surprisingly, below $T/K_0 \simeq 0.05$
we find strong indications of a second tricritical point where the first-order
transition becomes continuous. These results suggest an unexpected second-order
phase transition extending down to zero temperature, contrary to the prevailing
consensus. If confirmed, these findings reopen the question of the detailed
characterization of the transition including a suitable field theory
description.
