  Gottesman-Kitaev-Preskill (GKP) codes are a promising candidate for
implementing fault tolerant quantum computation in quantum harmonic oscillator
systems such as superconducting resonators, optical photons and trapped ions,
and in recent years theoretical and experimental evidence for their utility has
steadily grown. It is known that logical Clifford operations on GKP codes can
be implemented fault tolerantly using only Gaussian operations, and several
theoretical investigations have illuminated their general structure. In this
work, we explain how GKP Clifford gates arise as symplectic automorphisms of
the corresponding GKP lattice and show how they are identified with the mapping
class group of suitable genus $n$ surfaces. This correspondence introduces a
topological interpretation of fault tolerance for GKP codes and motivates the
connection between GKP codes (lattices), their Clifford gates, and algebraic
curves, which we explore in depth. For a single-mode GKP code, we identify the
space of all GKP codes with the moduli space of elliptic curves, given by the
three sphere with a trefoil knot removed, and explain how logical degrees of
freedom arise from the choice of a level structure on the corresponding curves.
We discuss how the implementation of Clifford gates corresponds to
homotopically nontrivial loops on the space of all GKP codes and show that the
modular Rademacher function describes a topological invariant for certain
Clifford gates implemented by such loops. Finally, we construct a universal
family of GKP codes and show how it gives rise to an explicit construction of
fiber bundle fault tolerance as proposed by Gottesman and Zhang for the GKP
code. On our path towards understanding this correspondence, we introduce a
general algebraic geometric perspective on GKP codes and their moduli spaces,
which uncovers a map towards many possible routes of future research.
