An inverse theorem for Gowers norms of trace functions over \(\mathbb F_p\) (Q2845624)

Let \(F = \mathbb F_p(T)\) be the field of rational functions with coefficients in the finite field \(\mathbb F_p\). Consider the Galois group \(G = \mathrm{Gal}(\bar{F}|F)\) of some separable closure of \(F\), and finite-dimensional representations \(\rho\colon G\rightarrow \mathrm{GL}(V)\). Let \(l \neq p\) be a prime number and let \(\iota\) be the isomorphism between the algebraic closure of the field of \(l\)-adic numbers and the field of complex numbers. The trace function is the function \(x\mapsto \iota(\mathrm{Tr}(\rho(\text{Fr}_x)\mid V^{I_x}))\), where \(\text{Fr}_x\) is the Frobenius automorphism at \(x\), and \(V^{I_x}\) denotes the subspace of \(V\) invariant under the action of \(I_x\), where \(I_x\) is the inertia group. In this paper the authors proved algebraic structure theorems for Gowers norms of trace functions. Their proofs based on some tools and concepts from algebraic geometry, and they used the Riemann Hypothesis over finite fields.