I am looking for an analogue of the following result of Fouvry and Katz for prime power modulus ("A general stratification theorem for exponential sums, and applications", J. reine angew. Math. 540 (2001), 115-166), which states:

Theorem (Fouvry-Katz): Let $n,d$ be positive integers and let $V$ be a locally-closed subscheme of $\mathbb{A}_{\mathbb{z}}^n$ with complex dimension at most $d$. Let $f(\mathbf{x})$ be an element in $\mathbb{Z}[x_1, \cdots, x_n]$.

Then there exists a number $C(n,d,f,V)$, closed subschemes $X_1, \cdots, X_n$ of relative dimension $n -j$ and $\mathbb{A}_{\mathbb{Z}}^n \supset X_1 \supset \cdots \supset X_n$ such that for any invertible function $g$ on $V$, any prime $p$, any $h \in \mathbb{A}^n(\mathbb{F}_p) \setminus X_j(\mathbb{F}_p)$ we have

$$\displaystyle \left \lvert \sum_{\mathbf{x} \in V(\mathbb{F}_p)} \chi(g(\mathbf{x})) \psi(f(\mathbf{x}) + h_1 x_1 + \cdots + h_n x_n) \right \rvert \leq C(n,d,f,V) p^{\frac{d}{2} + \frac{j-1}{2}}$$

for every non-trivial additive character $\psi$ of $\mathbb{F}_p$ and for every multiplicative character (possibly trivial) $\chi$ of $\mathbb{F}_p^\times$.

In particular, I ask whether there exists an analogous bound for exponential sums of the form

$$\displaystyle \sum_{\mathbf{x} \in V(\mathbb{Z}/p^2 \mathbb{Z})} \exp\left(2\pi i\left(h_1 x_1 + \cdots + h_n x_n \right)/p^2 \right)$$

of the quality as in the above theorem of Fouvry and Katz.


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