  One of the many remarkable properties of conformal field theory in two
dimensions is its connection to algebraic geometry. Since every compact Riemann
surface is a projective algebraic curve, many constructions of interest in
physics (which a priori depend on the analytic structure of the spacetime) can
be formulated in purely algebraic language. This opens the door to interesting
generalizations, obtained by taking another choice of field: for instance, the
$p$-adics. We generalize the AdS/CFT correspondence according to this
principle; the result is a formulation of holography in which the bulk geometry
is discrete---the Bruhat--Tits tree for $\mathrm{PGL}(2,\mathbb{Q}_p)$---but
the group of bulk isometries nonetheless agrees with that of boundary conformal
transformations and is not broken by discretization. We suggest that this forms
the natural geometric setting for tensor networks that have been proposed as
models of bulk reconstruction via quantum error correcting codes; in certain
cases, geodesics in the Bruhat--Tits tree reproduce those constructed using
quantum error correction. Other aspects of holography also hold: Standard
holographic results for massive free scalar fields in a fixed background carry
over to the tree, whose vertical direction can be interpreted as a
renormalization-group scale for modes in the boundary CFT. Higher-genus bulk
geometries (the BTZ black hole and its generalizations) can be understood
straightforwardly in our setting, and the Ryu-Takayanagi formula for the
entanglement entropy appears naturally.
