Prime Power Moduli

• The paper introduces a p-adic framework, employing logarithmic representations and Weyl differencing, to bound character sums for prime power moduli.
• It establishes nontrivial upper bounds with near square-root cancellation, improving previous results for squarefree moduli by optimizing parameters like j1 and j2.
• The developed methods have applications in analyzing quadratic congruences, subconvexity problems for L-functions, and equidistribution in arithmetic combinatorics.

A prime power modulus is an integer of the form $q = p^n$, where $p$ is a prime number and $n \geq 1$ is an integer. Research on character sums and exponential sums evaluated at polynomials modulo such moduli is central to analytic number theory, with deep connections to the distribution of quadratic forms, the behavior of L-functions, and applications in arithmetic combinatorics, Diophantine equations, and cryptography. The recent work (Baier et al., 15 Aug 2025) addresses the problem of estimating short Dirichlet character sums to prime power moduli, when these sums are evaluated at binary quadratic forms, extending and complementing previous estimates in the case of squarefree (in particular, prime) moduli.

1. Problem Formulation and Main Results

The focus is on estimating smoothed character sums of the form

$$S_Q(\Psi, \Phi, A, B, M, N; \chi) = \sum_{x, y \in \mathbb{Z}} \Psi\left(\frac{x-A}{M}\right) \Phi\left(\frac{y-B}{N}\right) \chi(Q(x, y)),$$

where $Q(X, Y)$ is a fixed quadratic form with coefficients in $\mathbb{Z}_p$ (subject to certain coprimality assumptions), $\chi$ is a primitive Dirichlet character modulo $q = p^n$ with $n$ large, and $\Psi$, $\Phi$ are Schwartz class functions of compact support. $A$, $B$, $M$, $N$ are parameters reflecting the location and scale of the "window" over which the sum is taken.

After appropriate change of variables—completing the square—the quadratic form is typically reduced to a diagonal form, e.g., $Q(x, y) = c(\alpha x^2 + y^2)$. The sum $S_Q$ can then be decomposed into two types of sums, one of the form

$$T(\Phi, \widetilde{B}, N, \beta; \chi) = \sum_{y \in \mathbb{Z},\ (y,p)=1} \Phi\left(\frac{y - \widetilde{B}}{N}\right) \chi(\beta + y^2),$$

which become the key objects of analysis.

The main results consist of nontrivial upper bounds for $S_Q$ and $T$ in ranges of $N$ with $q = p^n$ increasing. These bounds improve on previous results for squarefree (especially prime) moduli and approach the "square-root" cancellation exponent in certain regimes. The exponents are explicitly optimized in terms of parameters $j_1$ and $j_2$ that depend on the local $p$-adic properties of the phase.

2. Methodological Innovations: p-adic Techniques and Weyl Differencing

The paper introduces and systematically applies several tools adapted to the $p$-adic context:

• Transformation to Additive Characters: For a primitive character $\chi$ modulo $p^n$, a result (Proposition 1) ensures that $\chi(1+pt)$ can be represented as

$$\chi(1+pt) = \exp\left( \frac{a_0 \log_p(1+pt)}{p^n} \cdot 2\pi i \right)$$

with $a_0 \in \mathbb{Z}_p^\times$. This allows the translation of $\chi(\beta + y^2)$ into an additive exponential with a $p$-adic logarithmic phase.

• $p$-adic Weyl Differencing: Proposition 2 provides a $p$-adic analogue of the classical Weyl differencing, particularly effective for controlling sums with polynomial or $p$-adic analytic phases. For $H = p^k$,

$$\left| \sum_w \Phi\left(\frac{w-C}{X}\right)e\left(\frac{F(w)}{p^n}\right) \right|^2 \leq XH + H \sum_{0<|h|<2X/H} \left| \sum_w \Phi_h\left(\frac{w-C}{X}\right) e\left(\frac{F(w+p^k h) - F(w)}{p^n}\right) \right|,$$

with appropriate functions $\Phi_h$.

• Poisson Summation over Progressions: Proposition 3 expresses the sum over an arithmetic progression modulo $q$ using the Poisson summation formula:

$$\sum_{t \equiv r (q)} \Omega\left(\frac{t-C}{X}\right) = \frac{X}{q} \sum_{m \in \mathbb{Z}} \widehat{\Omega}\left(\frac{mX}{q}\right)e\left(\frac{m(r-C)}{q}\right).$$

• Estimation of Complete Exponential Sums: Proposition 4 (due to Cochrane-Liu-Zheng) supplies estimates for

$$S_\alpha(f, p^m) = \sum_{n \bmod p^m, n \equiv \alpha (p)} e(f(n)/p^m),$$

in terms of the multiplicity $\nu$ of critical points (zeroes of normalized derivatives) of $f$, giving

$$|S_\alpha(f, p^m)| \lesssim \nu \cdot p^{t/(\nu+1)} p^{m(1-1/(\nu+1))}.$$

The iterative use of $p$-adic Weyl shifts, sometimes repeated, ultimately reduces the degree of the phase, at which point Poisson summation and critical point analysis yield explicit bounds.

3. Local Analysis and Optimization of Critical Point Multiplicities

A significant technical aspect is the detailed $p$-adic local analysis of the phase function and its derivatives. Utilizing the explicit formula for $\chi$ in terms of the $p$-adic logarithm, one analyzes

$$F(w) = a_0 \log_p(1 + p u^{-1} g(w)),\quad g(w) = 2vw + p w^2,$$

computing $F'(w)$ and $F''(w)$ as rational functions in $w$. By partitioning the domain of $w$ into residue classes, one determines the maximum multiplicity $\nu$ for which $F^{(k)}(w) \equiv 0$ modulo appropriate powers of $p$. The parameters $j_1$ and $j_2$ correspond to bounds for $\nu+1$ for first and second derivatives, and in many typical cases $j_1 = 3$ suffices. For primes with $p \equiv \pm5 \bmod{12}$, further improvements yield $j_2=3$ as well.

The sharpness of the resulting bounds depends crucially on these local $p$-adic properties and the associated critical point analysis.

4. Main Bounds and Their Range of Validity

After smoothing and combining the analytic machinery, two principal sets of bounds are produced. For example, with $j_1 = 3$:

• For $N$ in the range $$q^{(j_1 - 1)/(2j_1 - 1)} \leq N \leq q^{(3j_1 - 2)/(2(2j_1 - 1))}$$,

$$S_{Q,i}(\dots) \ll M N^{1/2} q^{\frac{j_1 - 1}{2(2j_1 - 1)}}$$

where $M$, $N$ are summation parameters. A second bound, from a double Weyl shift, offers better exponents for smaller $N$ relative to $q$. The precise exponents simplify further when $j_2=3$. The analysis comprehensively details the transition between these regimes and the optimization of parameters to ensure best possible cancellation.

These estimates are compared directly with and improve upon the prior results of Heath-Brown for squarefree (prime) moduli, both by refining the range of $N$ and by achieving superior exponents in the explicit bounds.

5. Applications and Consequences

The estimates achieved in (Baier et al., 15 Aug 2025) have multiple applications:

• Quadratic Congruences: The bounds are applicable to the problem of counting small solutions to quadratic congruences modulo prime powers, extending the type of results obtained for squarefree moduli.
• Subconvexity Problems: Nontrivial bounds for character sums to prime power moduli evaluated at quadratic forms contribute to subconvexity problems for L-functions in the depth aspect and complement $q$-aspect subconvexity results (Milićević, 2014, Milićević et al., 2019).
• Analytic Number Theory and Equidistribution: These results expand the toolkit for analyzing polynomial value distributions and are relevant for understanding the distribution of points on quadratic varieties over residue class rings modulo $p^n$.
• Methodological Impact: The blend of $p$-adic techniques with classical Fourier-analytic and Weyl differencing methods serves as a template for approaching other $p$-adic exponential sum problems involving higher degree polynomials or forms.

6. Integration with Existing Literature and Comparative Edge

The work complements and extends the estimates of Heath-Brown for squarefree moduli by deploying $p$-adic analytic methods effective for higher multiplicities; it directly utilizes the machinery initiated by Milićević and collaborators to frame the exponential sums in terms of $p$-adic critical point analysis and iterative Weyl differencing. The techniques detailed in the paper clarify how the arithmetic of prime power moduli—particularly the non-cyclic nature of the multiplicative group, and the resulting local structure—affect the analysis of exponential sums, enabling sharper results than those possible for more general or squarefree moduli in some settings.

7. Concluding Synthesis and Directions

The paper establishes a set of nontrivial estimates for character sums $S_Q(\dots)$ modulo prime powers as evaluated at binary quadratic forms, driven by an overview of $p$-adic logarithmic representations, Weyl differencing, Poisson summation, and critical point analysis for $p$-adic functions. These results reinforce the power of $p$-adic analytic methods in bounding arithmetic exponential sums and open avenues for further investigation of polynomial value statistics and associated L-function problems in the depth aspect. Building on and refining the foundation established for squarefree moduli, these results advance the analytic theory of character sums in settings with high $p$-adic invariance and complexity.