  In this work, we initiate the study of proximity testing to Algebraic
Geometry (AG) codes. An AG code $C = C(\mathcal{X}, \mathcal{P}, D)$ over an
algebraic curve $\mathcal{X}$ is a vector space associated to evaluations on
$\mathcal{P}$ of functions in the Riemann-Roch space $L_\mathcal{X}(D)$. The
problem of testing proximity to an error-correcting code $C$ consists in
distinguishing between the case where an input word, given as an oracle,
belongs to $C$ and the one where it is far from every codeword of $C$. AG codes
are good candidates to construct short proof systems, but there exists no
efficient proximity tests for them. We aim to fill this gap.
  We construct an Interactive Oracle Proof of Proximity (IOPP) for some
families of AG codes by generalizing an IOPP for Reed-Solomon codes introduced
by Ben-Sasson, Bentov, Horesh and Riabzev, known as the FRI protocol. We
identify suitable requirements for designing efficient IOPP systems for AG
codes. Our approach relies on a neat decomposition of the Riemann-Roch space of
any invariant divisor under a group action on a curve into several explicit
Riemann-Roch spaces on the quotient curve. We provide sufficient conditions on
an AG code $C$ that allow to reduce a proximity testing problem for $C$ to a
membership problem for a significantly smaller code $C'$.
  As concrete instantiations, we study AG codes on Kummer curves and curves in
the Hermitian tower. The latter can be defined over polylogarithmic-size
alphabet. We specialize the generic AG-IOPP construction to reach linear prover
running time and logarithmic verification on Kummer curves, and quasilinear
prover time with polylogarithmic verification on the Hermitian tower.
