The Fourier Ratio of Elliptic, Hyperelliptic, and L-Function Families

We study the Fourier Ratio of arithmetic trace signals over finite fields. For every fixed non-isotrivial one-parameter elliptic family, the Frobenius trace signal has maximal-order Fourier Ratio, \[\FR(f_p)\asymp \sqrt p,\] so its Fourier mass is necessarily delocalized rather than concentrated on a sparse set of frequencies. Constant-\(j\) twist families exhibit the same square-root behavior unless the trace signal is degenerate, in which case the Fourier Ratio remains bounded. For \(d\)-parameter elliptic and hyperelliptic families we obtain general estimates and constructions realizing each order \(p^{r/2}\), \(1\le r\le d\), including the maximal order \(p^{d/2}\). We also establish analogous square-root estimates for several finite families of \(L\)-functions. These results give a quantitative pseudorandomness criterion for arithmetic trace signals in terms of spectral delocalization.

This page follows the manuscript closely. It is written as mathematical exposition rather than as a summary page.

1 Introduction

A longstanding tradition in analytic number theory is that completely deterministic arithmetic objects often behave, in measurable ways, like random sequences [Cramer1936, ErdosKac1940, Davenport1937, Chowla1965, Montgomery1973, RudnickSarnak1996, KatzSarnak1999, Sarnak2011, GreenTao2012]. Probabilistic models of the primes go back at least to Cramér, while cancellation phenomena for arithmetic functions have long suggested that functions such as the Möbius function should behave like sequences of independent random signs [Cramer1936, Davenport1937, Chowla1965]. Random sequences provide a natural benchmark for the absence of detectable structure insofar as they exhibit strong cancellation, small correlations with structured sequences, and statistical behavior that cannot easily be predicted from limited information. This appears again more recently in the modern study of the Möbius function. In 2011, Sarnak formulated the Möbius randomness principle, predicting that the Möbius function should be asymptotically orthogonal to deterministic zero-entropy systems [Sarnak2011]. In 2012, Green and Tao proved that the Möbius function is strongly asymptotically orthogonal to every polynomial nilsequence [GreenTao2012]. Thus an entirely deterministic arithmetic function has essentially no correlation with a remarkably broad class of structured deterministic sequences. Their result gives a strong sense in which the Möbius function exhibits a property characteristic of random data in that structured tests are unable to detect substantial correlation with it. We continue this by studying a different random-like feature of arithmetic data called spectral delocalization. Typically, random signals do not concentrate their Fourier mass on a small collection of frequencies; rather, their spectra tend to be spread broadly across the available frequency space.

Following the recent work in [AldalehEtAl], we study the Fourier Ratio, which has emerged as a particularly effective measure of spectral concentration [AldalehEtAl, BIN, IPY, IIPY, IP, IMW, IMWUncertainty]. In particular, the Fourier Ratio provides a natural way to quantify this random-like spectral delocalization. For a finite signal \(f\) with Fourier transform \(\widehat{f}\), the Fourier Ratio is defined by \[\operatorname{FR}(f) \ = \ \frac{\lVert\widehat{f}\rVert_1}{\lVert\widehat{f}\rVert_2}.\] Heuristically, this behaves like the square root of an effective Fourier support size, and so a small Fourier Ratio indicates that the Fourier mass of a signal is concentrated on relatively few frequencies, while a large Fourier Ratio indicates that the signal is spectrally delocalized. More precisely, if \(f\) has \(M\) available Fourier frequencies, then Cauchy-Schwarz yields \[\operatorname{FR}(f)^2 \ \leq \ \left|\operatorname{supp}\widehat{f}\right| \ \leq \ M.\] Consequently, a Fourier Ratio of order \(\sqrt{M}\) forces the Fourier transform to occupy a positive proportion of the entire frequency space. In this sense, maximal-order Fourier Ratio captures a concrete spectral feature commonly associated with random signals.

Thus the main question of this paper is whether arithmetic trace signals exhibit this same random-like spectral behavior. We show that they indeed do, and at the maximal possible scale at that. Let \(E/\mathbb{Q}(T)\) be a fixed one-parameter elliptic family, and for a prime \(p\) define the normalized Frobenius trace signal by \[f_p(t) \ = \ \frac{a_p(E_t)}{\sqrt{p}}.\] Our main result is that every fixed non-isotrivial one-parameter elliptic family satisfies \[\operatorname{FR}(f_p) \ \asymp \ \sqrt{p}.\] Since \(\sqrt{p}\) is the largest possible order for a signal on \(\mathbb{F}_p\), these Frobenius trace signals have maximally delocalized Fourier spectra. Their Fourier mass cannot be concentrated on a sparse collection of additive characters, but must instead be spread across a positive proportion of all available frequencies. Thus these arithmetic trace signals share a fundamental spectral property of random signals.

This also has a direct learning-theoretic interpretation. Random signals are difficult to learn from incomplete information because they contain little exploitable low-complexity structure. Spectral sparsity is one particularly useful form of such structure: when a signal is well represented by only a small number of Fourier frequencies, it can often be approximated, sampled, or recovered from substantially less information [AldalehEtAl, BIN, IPY, IIPY]. A maximal-order Fourier Ratio removes this advantage by forcing any accurate Fourier representation to use a large portion of the available spectrum. In this sense, the same spectral delocalization that makes a signal look random also makes it difficult to learn using sparse Fourier models, and the arithmetic trace signals studied in this paper exhibit this same spectral trait. This places our results naturally within the classical number-theoretic philosophy above of a deterministic arithmetic object sharing a concrete property with an appropriate random model.

The arithmetic source of this delocalization is itself classical; square-root cancellation estimates of Weil, Deligne, Katz, Michel, and others show that Frobenius trace functions have small correlation with additive characters [Weil1948, Del80, katz1980sommes, Michel1995]. From the traditional analytic number-theoretic perspective, such cancellation is already viewed as a manifestation of pseudorandom behavior. From the Fourier-Ratio perspective, these estimates say that no individual Fourier frequency can carry a substantial portion of the signal. Our results show that, when combined across the full frequency space, this cancellation forces maximal spectral delocalization.

The rest of the paper continues by studying the full extent of this phenomenon; we also determine the behavior of constant-\(j\) elliptic families, where nontrivial twist families again have Fourier Ratio of order \(\sqrt{p}\), while constant trace signals have Fourier Ratio of constant order, signifying degeneracy of the trace signal. For elliptic families with \(d\) parameters, the possible behavior becomes substantially richer. We obtain general bounds and construct families realizing every order \(p^{r/2}\) for \(1\leq r\leq d\), including the maximal order \(p^{d/2}\). We then extend the same framework to hyperelliptic families, obtaining analogous one and multi-parameter results. Finally, we apply the spectral formulation of the Fourier Ratio to finite families of \(L\)-functions, obtaining square-root Fourier Ratios for Dirichlet families and corresponding square-root behavior for families of cuspidal newforms. Thus the random-like spectral delocalization studied here is not confined to a single elliptic setting, but appears across elliptic curves, hyperelliptic curves, and classical families of \(L\)-functions.

2 Preliminaries

We first record the purely analytic Fourier inequalities used throughout our work. The upper bound for the Fourier Ratio depends only on the size of the frequency space, while a lower bound follows from an upper bound for the largest Fourier coefficient. We then establish the arithmetic estimates needed to apply these inequalities to elliptic trace signals.

2.1 Fourier Ratio Inequalities

The size of the frequency space immediately gives the following universal bounds.

Proof. By Parseval’s identity, \(\lVert\widehat f\rVert_2=\lVert f\rVert_{\ell^2(\mu_p)}\). Then, since \(\lVert V\rVert_2\leq\lVert V\rVert_1\) for every finite vector \(V\), we can write that \(\operatorname{FR}(f) \ \geq \ 1\). For the upper bound, there are exactly \(p^d\) Fourier coefficients, so Cauchy-Schwarz yields \[\lVert\widehat f\rVert_1 \ \leq \ \sqrt{p^d}\lVert\widehat f\rVert_2 \ = \ p^{d/2}\lVert\widehat f\rVert_2.\] Thus, dividing by \(\lVert\widehat f\rVert_2\) proves the bounds. ◻

The lower endpoint occurs when the Fourier transform is supported on a single frequency, while the upper endpoint is the largest value permitted by the size of the frequency space. Thus, to show that an arithmetic signal has Fourier Ratio of maximal order, it suffices to show that no individual Fourier coefficient carries too much of the signal’s \(L^2\) energy.

Proof. First, Parseval’s identity gives that \(\lVert\widehat f\rVert_2=\lVert f\rVert_{\ell^2(\mu_p)}\). by writing \[\lVert\widehat f\rVert_2^2 \ = \ \sum_m\left|\widehat f(m)\right|^2 \ \leq \ \lVert\widehat f\rVert_\infty\sum_m\left|\widehat f(m)\right| \ = \ \lVert\widehat f\rVert_\infty\lVert\widehat f\rVert_1.\] and dividing by \(\lVert\widehat f\rVert_\infty\lVert\widehat f\rVert_2\) we obtain the first inequality. The second follows immediately from the assumed bound on \(\lVert\widehat f\rVert_\infty\). ◻

This inequality allows us to avoid estimating the Fourier \(\ell^1\)-norm directly: once the \(\ell^2\)-norm of the signal is controlled, and the largest Fourier coefficient (\(\ell^\infty\)) is bounded, the Fourier Ratio follows immediately.

2.2 Estimates for Elliptic Trace Signals

We next consider the estimates for elliptic trace signals that will be used throughout the one-parameter case. These give us control of both the size of the trace signal and the size of its Fourier coefficients.

Proof. As the \(j\)-invariant is non-constant, we may use a result of [Michel1995] to obtain \[\sum_{t\in\mathbb{F}_p} a_p(E_t)^2 = p^2+O(p^{3/2}).\] Then, since \(f_p(t)=a_p(E_t)/\sqrt p\) and \(\mu_p\) assigns mass \(1/p\) to each point, \[\lVert f_p\rVert_{L^2(\mu_p)}^2 =\frac{1}{p^2}\sum_{t\in\mathbb{F}_p}a_p(E_t)^2 =1+O(p^{-1/2}),\] which is [eq:boundsest1]. [Michel1995] also provides the second estimate \[\sup_{m\in\mathbb{F}_p}\left|\sum_{t\in\mathbb{F}_p}a_p(E_t)e^{-2\pi i mt/p}\right| \ \ll_E \ p.\] By the definition of the probability-normalized Fourier coefficient, we have \[\widehat f_p(m) \ = \ \frac{1}{p\sqrt p}\sum_{t\in\mathbb{F}_p}a_p(E_t)e^{-2\pi i mt/p},\] hence \[\lVert\widehat f_p\rVert_\infty \ \ll \ p^{-1/2},\] which is [eq:boundsest2]. ◻

When the \(j\)-invariant is constant, we cannot directly apply the results of [Michel1995]. However, in this case, we have the variation in the trace signal instead comes from twists of a fixed elliptic curve. These traces can be expressed using multiplicative characters; thus, the required Fourier estimates follow from classical character-sum bounds.

Proof. Away from the fixed finite set of zeros and poles, we have \(\left|\chi_p(R(t))\right|=1\), so \[\lVert g_p\rVert_{\ell^2(\mu_p)}^2 \ = \ |c_p|^2(1+O_R(p^{-1})).\] Then, for each \(m \in \mathbb{F}_p\), the unnormalized sum defining \(\widehat g_p(m)\) is a mixed multiplicative and additive character-sum of fixed degree. Consequently, the standard Weil bound [Weil1948] gives \[\left|\sum_{t \in \mathbb{F}_p}\chi_p(R(t))e^{-2\pi i mt/p}\right| \ \ll_R \ \sqrt p,\] which includes the \(m = 0\) case because \(\chi_p(R(T))\) is nontrivial due to the conditions on \(R(T)\). Therefore, \(\lVert\widehat g_p\rVert_\infty \ll_R \left|c_p\right|p^{-1/2}\). Lemma 3 gives the lower bound \(\operatorname{FR}(g_p) \gg_R \sqrt p\), while Lemma 2 with \(d = 1\) gives the upper bound \(\operatorname{FR}(g_p) \leq \sqrt p\). Finally, for a fixed finite linear combination, the diagonal terms preserve the main \(L^2\) energy, and the cross terms are again bounded by Weil character-sums. Thus, the Fourier coefficients are bounded term-by-term by the same estimate; since the number of terms is fixed, the same asymptotic argument applies. ◻

Thus, although the non-constant and constant \(j\)-invariant cases require different arithmetic estimates, they both obtain the same lower bound for their \(\ell^2\)-norms. Since every Fourier coefficient is \(O(p^{-1/2})\), it follows that in either case, the Fourier ratio is on the order of \(\sqrt p\).

2.3 Analytic Transfer Bounds

The estimates above show that the Fourier Ratio can be controlled once we know the size of the signal and the size of its largest Fourier coefficient. Therefore, we use these two quantities to obtain general transfer bounds for the families considered below.

Proof. The upper bound comes directly from Lemma 2, since a signal on \(\mathbb{F}_p\) has only \(p\) Fourier coefficients. Thus \[\operatorname{FR}(f_p) \ \leq \ \sqrt p.\] For the lower bound, Lemma 3 gives \[\operatorname{FR}(f_p) \ \geq \ \frac{\lVert f_p\rVert_{\ell^2(\mu_p)}}{\lVert\widehat f_p\rVert_\infty}.\] Using the two assumptions therefore gives \(\operatorname{FR}(f_p) \ \gg \ \sqrt p\) and combining the two estimates gives what we want. ◻

Proof. The lower bound follows from Lemma 3, and the upper bound from Lemma 2. When \(r=d\), the two bounds have the same order, giving \(\operatorname{FR}(f_p)\asymp p^{d/2}\). ◻

The distinction between the one and multiple-parameter cases has now become clear; for \(d=1\), the estimate \(p^{-1/2}\) for the largest Fourier coefficient determines the ratio completely; however, for \(d>1\), the same estimate gives only the lower end of a larger possible range. The remaining question is then which values inside this larger range can actually occur.

When \(d=1\), an estimate \(\lVert\widehat f_p\rVert_\infty\ll p^{-1/2}\), together with \(\lVert f_p\rVert_{L^2(\mu_p)}\asymp1\), forces \(\operatorname{FR}(f_p)\asymp\sqrt p\). When \(d>1\), the same estimate yields only \(\operatorname{FR}(f_p)\gg\sqrt p\), while the universal upper bound is \(p^{d/2}\). It is therefore natural to ask which orders between these two extremes can actually occur for elliptic trace families.

3 Elliptic Families

3.1 One-Parameter Families

We first apply our transference theorem to elliptic families over a one-dimensional parameter space. When the j-invariant is non-constant, the result follows directly from Theorem 4. When the j-invariant is constant, the family is, after a suitable change of variables, a family of twists of a fixed elliptic curve, and a separate argument is required. In this case the trace may either exhibit nontrivial multiplicative-character variation or be constant on the smooth parameter locus.

3.1.1 Families With Non-Constant \(j\)-Invariant

Proof. By Theorem 4, \[\lVert f_p\rVert_{\ell^2(\mu_p)} \ \asymp_E \ 1 \qquad \text{and} \qquad \lVert\widehat f_p\rVert_\infty \ \ll_E \ p^{-1/2}.\] Thus, the hypotheses of Theorem 6 are satisfied, and hence, \[\operatorname{FR}(f_p) \ \asymp_E \ \sqrt p.\] Then our universal Cauchy-Schwarz bounds give the upper bound while the Fourier estimate from Theorem 4 gives the lower bound. ◻

Thus, every one-parameter family with non-constant j-invariant has Fourier Ratio of maximal order, \(\sqrt{p}\).

3.1.2 Families With Constant \(j\)-Invariant

A constant \(j\)-invariant does not by itself force the trace signal to be constant; rather, the family is described by twists of a fixed curve, so the parameter dependence instead appears through multiplicative characters. In the nontrivial cases, this character variation still produces the same square root Fourier behavior.

Proof. After excluding finitely many primes, a constant \(j\) elliptic family is formed by twists of a fixed elliptic curve. More precisely, if \(j\neq 0,\ 1728\), the twist parameter is given by a quadratic character. If \(j=1728\) or \(j=0\), the character has order dividing 4 or 6 respectively. See [Des18]. Thus, on the smooth parameter locus, the normalized trace is given by \[f_p(t) = \sum_{\nu=1}^{M}c_{\nu,p}\chi_{\nu,p}(R_\nu(t)),\] for a fixed rational function \(R\) and a nontrivial multiplicative character \(\chi_p\) of bounded order, with \(c_p\) depending only on \(p\). Since \(f_p\neq 0\), at least one coefficient \(c_{\nu,p}\) is nonzero. Lemma 5 gives \[\lVert f_p\rVert_{\ell^2(\mu_p)} \ \asymp_E \ \Bigl(\sum_\nu|c_{\nu,p}|^2\Bigr)^{1/2}\] and \[\lVert\widehat f_p\rVert_\infty \ \ll_E \ p^{-1/2}\Bigl(\sum_\nu|c_{\nu,p}|^2\Bigr)^{1/2}.\] The lower bound follows from Lemma 3 while Lemma 2 with \(d = 1\) gives the upper bound. Hence \(\operatorname{FR}(f_p) \ \asymp_E \ \sqrt p\) and changing finitely many singular values does not affect the order of either estimate. ◻

The remaining case is the concentrated case in which the normalized trace does not vary on the smooth parameter set at all. Instead, here the zero frequency contains essentially all of the Fourier mass, and the Fourier Ratio drops to the order of a constant.

Proof. Begin by writing \(f_p = c_p1_{U_p}\) with \(U_p\) as the smooth parameter set. Then \(\mathbb{F}_p\setminus U_p\) has \(O(1)\) points. If \(c_p = 0\), there is nothing to prove, and we are done. Otherwise, the Fourier Ratio is unchanged by multiplication by \(c_p\), so we can continue by merely considering \(1_{U_p}\). Starting with the zero frequency, the probability-space Fourier coefficient is \(\frac{|U_p|}{p} = 1 + O(p^{-1})\). For \(m\neq 0\), \[\frac{1}{p}\sum_{t\in U_p}e^{2\pi imt/p} \ = \ -\frac{1}{p}\sum_{t\not\in U_p}e^{2\pi imt/p}.\] Since \(\mathbb{F}_p\setminus U_p\) has \(O(1)\) elements, the nonzero Fourier coefficient is \(O(p^{-1})\). Hence \[\lVert\widehat{1_{U_p}}\rVert_1 \ = \ O(1) \qquad \text{and} \qquad \lVert\widehat{1_{U_p}}\rVert_2 \ \asymp_E \ 1.\] Therefore, \(\operatorname{FR}(f_p) \ \asymp_E \ 1\). If \(U_p \ = \ \mathbb{F}_p\), then the signal is constant, and its Fourier transform is supported only at the zero frequency, hence giving \(\operatorname{FR}(f_p) \ = \ 1\). ◻

Together, Theorems 8 and 10 describe the two behaviors that appear in the one-parameter setting considered here; nontrivial oscillation forces maximal order Fourier Ratio while a constant trace gives the oppposite behavior.

3.2 Multiple Parameter Families

For a signal on \(\mathbb{F}_p^d\), the number of available frequencies is \(p^d\), so our universal upper bound becomes \(p^{d/2}\). However, a square root saving in an additive twist still gives only the lower bound \(\sqrt p\) from Lemma 3 which leaves a substantially larger range of possible Fourier Ratios when \(d>1\).

3.2.1 A Universal Upper Bound

Proof. The signal is just a function on \(\mathbb{F}_p^d\), so the result is precisely Lemma 2. ◻

Unlike the one-parameter case, the lower bound \(\sqrt p\) need not match the universal upper bound once \(d>1\), so the Fourier Ratio is no longer determined by square root cancellation alone.

3.2.2 Bounds From Elliptic Trace Estimates

Proof. By the additive twist estimate and the normalization of the Fourier transform, \[\lVert\widehat f_p\rVert_\infty \ \ll_E \ p^{-1/2}.\] Applying Theorem 7 with \(r \ = \ 1\) gives the lower bound \(\operatorname{FR}(f_p) \ \gg_E \ \sqrt p\), and Theorem 11 gives the upper bound \(\operatorname{FR}(f_p) \ \ll_d \ p^{d/2}\). ◻

This theorem deliberately keeps the two bounds separate because without stronger cancellation in the \(d\)-dimensional Fourier transform, there is no reason for the lower bound to rise to \(p^{d/2}\). We see this in the coming examples.

3.2.3 Sharpness of Bounds

Proof. For the first family, using the geometric trace, the character-sum identity \[f_p(a,b) \ = \ -p^{-1/2}\sum_x\chi(x^3+ax+b)\] holds exactly for all \((a,b) \in \mathbb{F}_p^2\), including the singular fibers. For \((m, n) \in \mathbb{F}_p^2\), write \[\widehat{f}_p(m,n) = \frac{1}{p^2} \sum_{a,b \in \mathbb{F}_p} f_p(a,b)e^{-2\pi i(ma+nb)/p}.\]

If \(n = 0\), summing first over \(b\) gives zero. If \(n \neq 0\), then \[\sum_{b \in \mathbb{F}_p} \chi(x^3 + ax + b)e^{-2\pi i nb/p} \ = \ G(\chi) e^{2\pi i n(x^3+ax)/p},\] where \(|G(\chi)| = \sqrt{p}\). The remaining sum over \(a\) is \[\sum_{a \in \mathbb{F}_p} e^{2\pi i(nx-m)a/p},\] which vanishes unless \(nx = m\), in which case it has magnitude \(p\). Thus \[|\widehat{f}_p(m,n)| \ = \ 1\] for every \(n \neq 0\), with \(x = m/n\), and it is zero for \(n = 0\). Thus, \(\left|\widehat f_p(m,n)\right|=1\) for every pair with \(n\neq 0\), giving exactly \(p(p-1)\) nonzero Fourier coefficients of magnitude 1. Thus, \[\operatorname{FR}(f_p) \ = \ \sqrt{p(p-1)} \ = \ p + O(1).\] For the second statement in the proposition, we have scalar multiplication does not change the Fourier Ratio, and further that the Fourier Ratio is multiplicative under tensor products. Hence, since \(\operatorname{FR}(1) = 1\) and the quadratic Gauss sum gives \(\operatorname{FR}(\chi) = \sqrt{p-1}\), we obtain \[\operatorname{FR}(g_p) \ = \ (p-1)^{r/2}.\] ◻

This proposition shows that, in higher dimensions, the Fourier Ratio can reflect the number of parameter directions in which the signal oscillates. In particular, without additional hypotheses, one can obtain orders ranging from \(p^{1/2}\) up to the maximal value \(p^{d/2}\).

4 Hyperelliptic Families

The preceding arguments suggest that the same Fourier-Ratio behavior should persist for more general families of curves whenever their trace functions satisfy analogous square-root cancellation estimates. We therefore consider one-parameter families of hyperelliptic curves. The Fourier-analytic argument is unchanged; the main new input is a uniform square-root bound for the additive twists of the normalized hyperelliptic trace function.

4.1 One-Parameter Families

Proof. We follow the proof outlined in the elliptic case in [Michel1995]. We start by fixing a prime \(\ell\). Fix an isomorphism \(\iota: \overline{\mathbb{Q}}_\ell \xrightarrow{\sim} \mathbb{C}\). Define our one-parameter hyperelliptic curve family, \(C_t\), over \(\mathbb{Z}\), that is, the scheme over \(\mathbb{Z}\) given by \(f:V\to \mathbb{A}^1_{\mathbb{Z}[1/6\ell]}\).

To study the fiber \(f^{-1}(t)\) for some \(t\) and to describe this situation, we consider the sheaf \[\mathcal{F}\ := \ R^1f_*\overline{\mathbb{Q}}_\ell.\]

Note that this returns \(\mathcal{F}_t \ \cong \ H^1(C_t)\). We can also find an open subset \(U\) as in [Michel1995] such that \(\mathcal{F}\) is lisse of rank \(2g\) on \(U\).

Note that \[\begin{aligned} \text{Tr}(\text{Frob}_{p,t} \mid \mathcal{F}_p) \ = \ a_{p,t}. \end{aligned}\] We take the Tate twist by 1/2 with the rank-1 Weil sheaf \(\overline{\mathbb{Q}}_\ell(1/2)\) to get \[\text{Tr}(\text{Frob}_{p,t} \mid \mathcal{F}_{p}(1/2)) \ = \ \frac{a_{p,t}}{\sqrt{p}}.\] The same relation holds in the hyperelliptic case since \[\text{Tr}(\text{Frob}_{p,t} \mid \mathcal{F}_p(1/2)) \ = \ \text{Tr}\left(\text{Frob}_{p,t} \mid \mathcal{F}_p \ \otimes \ \overline{\mathbb{Q}}_\ell(1/2)\right) \ = \ \text{Tr}(\text{Frob}_{p,t} \mid \mathcal{F}_p)\text{Tr}\left(\text{Frob}_{p,t} \mid \overline{\mathbb{Q}}_\ell(1/2)\right) \ = \ \frac{a_{p,t}}{\sqrt{p}}.\] When \(\psi_p\) is non-trivial, we can define the wildly ramified sheaf \(\mathcal{L}_{\psi_p}\). This is defined in both [Michel1995] and [katz1980sommes].

Then, applying the Lefschetz-Grothendieck trace formula, when \(\psi_p\) is non-trivial, we have \[\sum_{t\in \mathbb{F}_p, \Delta(t)\neq0}\text{Tr}(\text{Frob}_{p,t} \mid \mathcal{F}_{p}(1/2))\psi_p (t) \ = \ \sum_{i=0}^{2}(-1)^{i}\text{Tr}(\text{Frob}_{p} \mid H_{c}^{i}(\overline{U}_{p},\mathcal{F}_{p}(1/2)\otimes\mathcal{L}_{\psi_{p}}))\] where \(\overline{U_p}\) is the fiber of \(U\), the open subset where \(\mathcal{F}\) is defined. See [KatzSarnak1999] for more details.

We next use that the representation corresponding to \(\mathcal{F}_p\) is irreducible [KatzSarnak1999]. Let us denote this representation by \(\rho_0\). Then, the representation corresponding to \(\mathcal{F}_p(1/2)\) is given by \(\rho \ := \ \rho_0 \ \otimes \ \chi_{1/2}\) where \(\chi_{1/2}\) is the representation corresponding to the rank-1 Weil sheaf \(\overline{\mathbb{Q}}_\ell(1/2)\). Since \(\rho_0\) is an irreducible representation of dimension \(2g\) and \(\chi_{1/2}\) is 1-dimensional, \(\rho\) remains an irreducible representation of dimension \(2g\). In particular, \(\rho\) is an irreducible representation of a subgroup of \(\text{Sp}(2g)\), which is the monodromy group for this family of hyperelliptic curves since we have fuill geometric monodromy. We also have that \(\mathcal{F}_p\) is tamely ramified at \(\infty\) by Section 10 of [KatzSarnak1999]. This property is preserved by the twist, so \(\mathcal{F}_p(1/2)\) is also tamely ramified. Further, \(\mathcal{F}_p\) is constructible over the \(\ell\)-adics. We can therefore apply the key lemma in Section 4.8 of [katz1980sommes] to obtain \[H_{c}^{2}(\overline{U}_{p},\mathcal{F}_{p}(1/2) \ \otimes \ \mathcal{L}_{\psi_{p}}) \ = \ 0.\] Since \(U\) is affine, we also have \[H_{c}^{0}(\overline{U}_{p},\mathcal{F}_{p}(1/2) \ \otimes \ \mathcal{L}_{\psi_{p}}) \ = \ 0.\]

\(\mathcal{F}_p\) is pure and has weight 1 [KatzSarnak1999, Del80], while \(\overline{\mathbb{Q}}_\ell(1/2)\) has weight \(-1\), so \(\mathcal{F}_p(1/2)\) is pure and has weight 0. Therefore, \(\mathcal{F}_p(1/2)\otimes \mathcal{L}_{\psi_p}\) is also pure of weight 0, and the eigenvalues of the first cohomology group satisfy the same bound. It follows that \[\left| \sum_{t\in \mathbb{F}_p, \Delta(t)\neq 0}\text{Tr}(\text{Frob}_{p,t} \mid \mathcal{F}_{p}(1/2))\psi_p(t) \right| \ \leq \ D_h \sqrt{p}\] where \(D_h \ = \ \dim(H_{c}^{1}(\overline{U}_{p},\mathcal{F}_{p}(1/2) \ \otimes \ \mathcal{L}_{\psi_{p}}))\). See [freitag1988etale] for more details. all that remains is to bound the degenerate terms. The traces are bounded by \(2g\), and the number of singular parameters gives an \(O(1)\) contribution. Hence, even for hyperelliptic curves we obtain \[\left|\sum_{t\in \mathbb{F}_p}\frac{a_{p,t}}{\sqrt{p}}\psi_p(t)\right| \ = \ O(\sqrt{p}).\] ◻

The previous theorem gives the additive character estimate that replaces the corresponding elliptic estimate used earlier in the paper. We continue by combining this with the second moment estimate for the hyperelliptic trace function to control both the size of the signal and its largest Fourier coefficient.

Proof. First, using the second moment estimate for the hyperelliptic trace function [KatzSarnak1999, Del80], we have \[\begin{aligned} \|f\|_{\ell^2(\mu_p)}^2 \ &= \ \frac{1}{p}\sum_{t\in\mathbb{F}_p}|f(t)|^2 \\ \ &= \ \frac{1}{p}\sum_{t\in\mathbb{F}_p}\frac{a_p(C_t)^2}{p} \\ \ &= \ 1+O(p^{-1/2}), \end{aligned}\] which gives the first estimate. For our second estimate, using the probability-normalized Fourier transform from Definition 1, we have \[\begin{aligned} \|\widehat{f}\|_\infty \ &= \ \max_{m\in\mathbb{F}_p}\left|\frac{1}{p}\sum_{t\in\mathbb{F}_p}f(t)\chi(-mt)\right| \\ \ &= \ \frac{1}{p}\max_{m\in\mathbb{F}_p}\left|\sum_{t\in\mathbb{F}_p}\frac{a_p(C_t)}{p^{1/2}}\chi(-mt)\right|. \end{aligned}\] For \(m \neq 0\), \(\chi(-mt)\) is a non-trivial additive character, so Theorem 14 gives \[\left|\sum_{t\in\mathbb{F}_p}\frac{a_p(C_t)}{p^{1/2}}\chi(-mt)\right|\ = \ O(\sqrt{p}).\] At \(m = 0\), geometric nontriviality gives the corresponding trivial character estimate [Del80] \[\left|\sum_{t\in\mathbb{F}_p}\frac{a_p(C_t)}{p^{1/2}}\right|\ = \ O(\sqrt{p}).\] Hence \(\|\widehat{f}\|_\infty = O(p^{-1/2})\). ◻

At this point we have exactly the two estimates needed for the Fourier Ratio argument: control of the \(L^2\) energy of the trace signal and a uniform bound for its Fourier coefficients. Thus, we combine them in the same way as in the earlier elliptic setting.

Proof. The proof is the same Fourier-ratio argument used in the elliptic case. The preceding estimates give \[\|\widehat f\|_2^2=1+O(p^{-1/2}) \qquad\text{and}\qquad \|\widehat f\|_\infty=O(p^{-1/2}).\] Hence \[\|\widehat f\|_2^2\leq \|\widehat f\|_\infty\|\widehat f\|_1\] implies \(\|\widehat f\|_1\gg p^{1/2}\), while Cauchy–Schwarz gives \(\|\widehat f\|_1\ll p^{1/2}\). Since \(\|\widehat f\|_2\asymp1\), we obtain \(\operatorname{FR}(f)\asymp p^{1/2}\). ◻

Therefore, we obtain the same asymptotic Fourier Ratio for the hyperelliptic families considered here as in the one-parameter elliptic setting, and although the arithmetic input is different, the same spectral behavior follows once the required trace and Fourier estimates can be established.

4.2 Multiple Parameter Families

We can now extend our previous results to hyperelliptic families over \(\mathbb{F}_p^d\). Just as in the elliptic setting, the universal upper bound becomes \(p^{d/2}\) while the lower bound depends on the amount of cancellation available in the additive twists. Thus, the one-parameter result does not immediately determine the Fourier Ratio once \(d > 1\).

4.2.1 A Universal Upper Bound

Proof. The signal is simply a function on \(\mathbb{F}_p^d\), so this follows immediately from Lemma 2. ◻

As before, this bound depends only on the number of available frequencies and therefore requires no additional information about the hyperelliptic family.

4.2.2 Bounds from Hyperelliptic Trace Estimates

Proof. Dividing the second estimate by \(p^d\) gives \[\lVert\widehat f_p\rVert_\infty \ \ll_C \ p^{-1/2}.\] Since \(\lVert f_p\rVert_{\ell^2(\mu_p)}\asymp_C1\), Theorem 7 with \(r = 1\) gives \(\operatorname{FR}(f_p) \gg_C \sqrt p\). The upper bound follows from Theorem 17, giving \[\sqrt p \ \ll_C \ \operatorname{FR}(f_p) \ \ll_d \ p^{d/2}.\] ◻

Thus, exactly as for elliptic curves, square root cancellation in the additive twists gives only the lower bound \(\sqrt p\) in several parameters. Stronger cancellation can increase this lower bound; more generally, if \[\lVert\widehat f_p\rVert_\infty \ \ll_C \ p^{-r/2}\] for some \(1 \leq r \leq d\), then Theorem 7 gives \(\operatorname{FR}(f_p) \gg_C p^{r/2}\), and when \(r = d\) we obtain the maximal order \(\operatorname{FR}(f_p) \asymp_C p^{d/2}\).

4.2.3 Sharpness of Bounds

Proof. Whenever \(t_1\cdots t_r \neq 0\), the fiber \(C_{\mathbf t}\) is the quadratic twist of \(C_0\) by \(\prod_{j=1}^rt_j\). Therefore, if \(\chi\) is the quadratic character of \(\mathbb{F}_p\), \[a_p(C_{\mathbf t}) \ = \ a_p(C_0)\chi\left(\prod_{j=1}^{r}t_j\right) \ = \ a_p(C_0)\prod_{j=1}^{r}\chi(t_j).\] Taking \(\chi(0) = 0\) gives the same identity after our zero extension and thus \[f_p(\mathbf t) \ = \ \frac{a_p(C_0)}{\sqrt p}\prod_{j=1}^{r}\chi(t_j).\] As already established, scalar multiplication does not change the Fourier Ratio and the remaining \(d-r\) variables are constant. Further, the Fourier Ratio is multiplicative under tensor products and \(\operatorname{FR}(\chi) = \sqrt{p-1}\). Thus \[\operatorname{FR}(f_p) \ = \ (p-1)^{r/2}.\] Taking \(r = d\) gives \(\operatorname{FR}(f_p) \asymp p^{d/2}\), so the universal upper bound is sharp, while varying \(r\) realizes every intermediate order. ◻

Thus, the explicit hyperelliptic families constructed above realize the same range of Fourier-Ratio orders as the corresponding elliptic examples. A family may have a Fourier Ratio as small as order \(\sqrt p\) while still exhibiting nontrivial oscillation, or it may use all \(d\) parameter directions and reach the maximal order \(p^{d/2}\). More generally, the examples above realize every order \(p^{r/2}\) between these two extremes.

5 L-Function Families

The previous sections study Fourier transforms of trace signals as the parameter of a geometric family varies over a finite field. However, if we instead use the spectral formulation of the Fourier Ratio developed in [FractalFourier2026], then we can ask an analogous question for families of \(L\)-functions. In particular, we fix a prime \(p\) and consider the normalized local coefficient as the members of a family vary, which for the cuspidal newforms considered below is the normalized Hecke eigenvalue \(\lambda_f(p)\). The vector of these values then plays the role of the Fourier vectors considered earlier, and thus for a family of \(L\)-functions of size \(D\), the Fourier Ratio again has the natural upper scale \(\sqrt{D}\). Since the general finite-dimensional bounds and the equidistribution result that converts a limiting coefficient distribution into an asymptotic Fourier Ratio are already proved in [FractalFourier2026], we use those results here and focus on the arithmetic input needed for the \(L\)-function families.

5.1 Dirichlet L-Functions

For Dirichlet \(L\)-functions, the local coefficients all have the same absolute value, hence the Fourier Ratio can be calculated without using an equidistribution argument. Consequently, this gives the simplest example of the local Fourier Ratio and already attains the natural upper bound.

Proof. For a Dirichlet character \(\chi\) modulo \(q\), the normalized local coefficient at \(p\nmid q\) is \[\lambda_\chi(p) \ = \ \chi(p), \qquad \left|\chi(p)\right| \ = \ 1.\] There are \(\varphi(q)\) characters modulo \(q\), so \[\lVert\Lambda_p(\mathcal{X}(q))\rVert_1 \ = \ \varphi(q), \qquad \lVert\Lambda_p(\mathcal{X}(q))\rVert_2 \ = \ \sqrt{\varphi(q)},\] which gives the first result. Every character modulo \(q\) is induced by a unique primitive character of conductor \(d\mid q\), so \[\varphi(q) \ = \ \sum_{d\mid q}\varphi^{*}(d),\] and Möbius inversion gives \[\varphi^{*}(q) \ = \ \sum_{d\mid q}\mu(q/d)\varphi(d).\] The same norm calculation gives \(\operatorname{FR}_p(\mathcal{X}^{*}(q))=\sqrt{\varphi^{*}(q)}\), while for odd prime \(q\) we have \(\varphi^{*}(q)=q-2\). ◻

The same calculation continues to hold true even if we begin with a fixed cuspidal form and vary only the Dirichlet character. Twisting merely multiplies the local Hecke eigenvalue by \(\chi(p)\), which has absolute value one, so every coefficient again has the same magnitude.

Proof. Since \[\lambda_{f\otimes\chi}(p) \ = \ \lambda_f(p)\chi(p)\] and \(\left|\chi(p)\right|=1\), every entry of \(\Lambda_p(\mathcal{F}_{f,\mathcal{X}})\) has absolute value \(\left|\lambda_f(p)\right|\). Therefore \[\lVert\Lambda_p(\mathcal{F}_{f,\mathcal{X}})\rVert_1 \ = \ \#\mathcal{X}\left|\lambda_f(p)\right|, \qquad \lVert\Lambda_p(\mathcal{F}_{f,\mathcal{X}})\rVert_2 \ = \ \sqrt{\#\mathcal{X}}\left|\lambda_f(p)\right|,\] and the result follows. ◻

5.2 Cuspidal Newform L-Functions

For a family of cuspidal newforms the local coefficients no longer have constant absolute value, so the calculation we used above has become inaccessible. Instead, we must use the fact that their distribution is known as the family grows.

Let \(H_k^*(N)\) denote the set of cuspidal newforms of level \(N\) and weight \(k\), and let \[\mathcal{F}_N \ = \ \{L(s,f):f\in H_k^*(N)\}.\] For \(f\in H_k^*(N)\), let \(\lambda_f(n)\) denote the normalized Hecke eigenvalues, and write \[D_{N,k} \ = \ \#H_k^*(N).\]

For the cuspidal newform families, this coefficient vector has a natural spectral interpretation, if w let \(T_p\) denote the Hecke operator at \(p\) and then define the normalized Hecke operator as \[\widetilde{T}_p \ := \ p^{-(k-1)/2}T_p.\] Following suit, since every \(f\in H_k^*(N)\) is a Hecke eigenform we have \[\widetilde{T}_p f \ = \ \lambda_f(p)f.\] Therefore \[\Lambda_p(\mathcal{F}_N) \ = \ \bigl(\lambda_f(p)\bigr)_{f\in H_k^*(N)}\] is merely the spectrum of \(\widetilde{T}_p\) counted with multiplicity across the family. Thus, as in the spectral formulation of [FractalFourier2026], the natural Hecke operator itself may become the vector whose Fourier Ratio we study. With this as our setting, we can use Serre’s vertical equidistribution theorem which describes the limiting distribution of this spectrum, allowing the finite-family spectral result of [FractalFourier2026] to determine its Fourier Ratio. fix a prime \(p\nmid N\). Serre’s vertical equidistribution theorem [SerreHecke1997] shows that, as \(N+k\to\infty\) through \(p\nmid N\) with \(D_{N,k}\to\infty\), the values \(\lambda_f(p)\) become equidistributed on \([-2,2]\) with respect to the measure \[d\mu_p(x) \ = \ \frac{p+1}{\pi} \frac{\sqrt{1-x^2/4}} {(\sqrt p+p^{-1/2})^2-x^2}\,dx.\]

Since the distribution of the entries of \(\Lambda_p(\mathcal{F}_N)\) is known, the finite-family result of [FractalFourier2026] allows us to directly turn this into an asymptotic formula for the Fourier Ratio. In particular, the first absolute moment of \(\mu_p\) determines the \(\ell^1\) norm while its second moment determines the \(\ell^2\) norm.

Proof. Applying the finite-family equidistribution theorem of [FractalFourier2026] to the distribution above gives \[\operatorname{FR}_p(\mathcal{F}_N) \ = \ \sqrt{D_{N,k}} \frac{\displaystyle\int_{-2}^{2}\left|x\right| \ d\mu_p(x)}{\displaystyle\left(\int_{-2}^{2}x^2 \ d\mu_p(x)\right)^{1/2}}+o(\sqrt{D_{N,k}}).\] Using Serre’s moment formula with \(X_2(x)=x^2-1\) gives \[\int_{-2}^{2}x^2 \ d\mu_p(x) \ = \ 1+p^{-1}.\] Also, by symmetry and the substitutions \(u=1-x^2/4\) and then \(u=t^2\), \[\int_{-2}^{2}\left|x\right| \ d\mu_p(x) \ = \ \frac{2(p+1)}{\pi} \left(1-\frac{p-1}{\sqrt p}\arctan\frac{1}{\sqrt p}\right).\] Substituting these two expressions into the equidistribution formula gives the stated value of \(C_p\), and since \(C_p>0\) we obtain \(\operatorname{FR}_p(\mathcal{F}_N)\asymp_p\sqrt{D_{N,k}}\). ◻

The same argument can be further extended to newforms with a fixed nebentypus. In this case, the character introduces a phase into the local Hecke eigenvalue, but after removing the added phase, the normalized eigenvalues have the same limiting distribution \(\mu_p\), thus the Fourier Ratio has the same asymptotic constant.

Proof. Choose a square root of \(\chi(p)\) and define \[\widetilde{\lambda}_f(p) \ = \ \chi(p)^{-1/2}\lambda_f(p).\] By [RossHecke2026], these normalized eigenvalues are equidistributed with respect to the same measure \(\mu_p\). Since \(\left|\chi(p)^{-1/2}\right|=1\), multiplying every entry of the coefficient vector by \(\chi(p)^{-1/2}\) does not change its Fourier Ratio. Thus, the result follows immediately from the finite-family equidistribution theorem of [FractalFourier2026], with the same constant \(C_p\) as in Theorem 23. ◻

The results above describe the Fourier Ratio in terms of the size of \(D_{N,k}\) of the relevant family. However, if the level itself is the parameter of interest, then a dimension asymptotic easily converts this into a statement in terms of \(N\).

Proof. For trivial nebentypus and prime level \(N\), \[D_{N,k} \ = \ \frac{k-1}{12}(N-1)+O_{k,\varepsilon}(N^{1/2+\varepsilon}).\] Substituting this into Theorem 23 gives the result. ◻

The local coefficient at \(p\) also determines the coefficients at higher powers \(p^m\) through the Hecke recurrence. Therefore, the distribution of \(\lambda_f(p)\) already contains the information we need so as to study the Fourier Ratio at every fixed prime power.

Proof. The Hecke recurrence gives \[\lambda_f(p^m) \ = \ X_m(\lambda_f(p)).\] Therefore the finite-family equidistribution theorem of [FractalFourier2026], now applied with \(h=X_m\), gives \[\operatorname{FR}_{p^m}(\mathcal{F}_N) \ = \ \sqrt{D_{N,k}}\frac{\displaystyle\int_{-2}^{2}\left|X_m(x)\right| \ d\mu_p(x)}{\displaystyle\left(\int_{-2}^{2}X_m(x)^2 \ d\mu_p(x)\right)^{1/2}}+o(\sqrt{D_{N,k}}).\] The identity \[X_m(x)^2 \ = \ \sum_{j=0}^{m}X_{2j}(x)\] together with Serre’s moment formula [SerreHecke1997], which gives \(\int_{-2}^{2}X_{2j}(x)\,d\mu_p(x)=p^{-j}\), yields \[\int_{-2}^{2}X_m(x)^2 \ d\mu_p(x) \ = \ \sum_{j=0}^{m}p^{-j}.\] This gives the stated value of \(C_{p,m}\), while \(C_{p,m}>0\) because \(X_m\) is not identically zero and \(\mu_p\) has positive density on \((-2,2)\). ◻

Thus, once we know that the local Hecke eigenvalues are equidistributed with respect to some limiting measure, the Fourier Ratio immediately follows from said distribution. For the newform families considered above, the main arithmetic input was therefore the vertical equidistribution of the Hecke eigenvalues, while the resulting \(\sqrt{D}\) behavior is an immediate consequence of the finite-family theorem from [FractalFourier2026].

Acknowledgments

This research was supported with funding from the National Science Foundation (grant DMS2341670), Missouri State University, the University of Chicago, the University of Michigan, and Williams College. The authors are also grateful for the support from Texas A&M University, the University of Maryland, the University of Rochester, and Virginia Tech University.

The Fourier Ratio of Elliptic, Hyperelliptic, and \(L\)-Function Families

Abstract

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