Fixed-degree Atkin–Lehner sign statistics for Hilbert newforms

Abstract

We conjecture joint equidistribution of prescribed local Atkin–Lehner signs among genuine parallel-weight-two Hilbert newform orbits of a fixed Hecke-field degree. Ambient Plancherel equidistribution gives the corresponding statement without the rationality-degree restriction. The fixed-degree condition is global and thin, and no known trace-formula projector isolates it.

The fixed-degree family

Let $F$ be a real quadratic field with narrow class number one, let $d\ge1$, and let $S$ be a finite nonempty set of prime ideals of $\mathcal O_F$ satisfying $\mathfrak q\nmid2D_F$ for every $\mathfrak q\in S$. Let $\mathcal H_{F,d,S}(X)$ be the Hecke-Galois orbits $O$ of parallel-weight-two Hilbert newforms such that

• the central character is trivial;

• the form is non-CM and not a base change from $\mathbb Q$;

• $[K_O:\mathbb Q]=d$;

• $\mathrm N\mathfrak n_O\le X$;

• $v_{\mathfrak q}(\mathfrak n_O)=1$ for every $\mathfrak q\in S$.

Write $W_{\mathfrak q}(O)\in\{\pm1\}$ for the local Atkin–Lehner eigenvalue.

Conjecture 1. If $\#\mathcal H_{F,d,S}(X)\to\infty$, then for every $\eta=(\eta_{\mathfrak q})_{\mathfrak q\in S}\in\{\pm1\}^S$,

$$\lim_{X\to\infty} \frac{\#\{O\in\mathcal H_{F,d,S}(X):W_{\mathfrak q}(O)=\eta_{\mathfrak q} \text{ for all }\mathfrak q\in S\}} {\#\mathcal H_{F,d,S}(X)} =2^{-|S|}.$$

Numerical evidence

The available data contain $75268$ genuine orbits with $109751$ reliable local observations at primes of exact level valuation one. The largest complete singleton family is

$$F=\mathbb Q(\sqrt2),\qquad d=1,$$

at the inert prime of norm $9$: the two signs occur $604$ and $608$ times. Over $\mathbb Q(\sqrt5)$, the analogous degree-one count is $334$ versus $304$.

Several two-prime samples are shown in Table 1; columns correspond to the four sign vectors in a fixed order.

family	$++$	$+-$	$-+$	$--$
$\mathbb Q(\sqrt5)$, selected primes of norms $4$ and $5$, all degrees	$141$	$163$	$164$	$165$
$\mathbb Q(\sqrt5)$, same pair, degree $1$	$59$	$77$	$84$	$73$
$\mathbb Q(\sqrt5)$, primes of norms $4$ and $11$	$87$	$76$	$82$	$89$
$\mathbb Q(\sqrt5)$, primes of norms $5$ and $11$	$64$	$65$	$59$	$64$
$\mathbb Q(\sqrt5)$, latter pair, degree $1$	$29$	$31$	$31$	$29$
$\mathbb Q(\sqrt2)$, degree $1$, inert norms $25$ and $9$	$12$	$16$	$20$	$12$

Selected joint local-sign counts.

The finite samples are compatible with uniformity, but meaningful tests for larger sets $S$, large fixed degrees, and many base fields are not yet available.

Fourier reformulation

Put

$$N_d(X)=\#\mathcal H_{F,d,S}(X)$$

and, for $T\subseteq S$, define the sign correlation

$$C_T(X)=\sum_{O\in\mathcal H_{F,d,S}(X)} \prod_{\mathfrak q\in T}W_{\mathfrak q}(O).$$

The identity

$$\mathbf 1_{\{W(O)=\eta\}} =2^{-|S|}\sum_{T\subseteq S} \prod_{\mathfrak q\in T}\eta_{\mathfrak q}W_{\mathfrak q}(O)$$

shows that Conjecture 1 is equivalent to

$$C_T(X)=o(N_d(X)) \qquad\text{for every nonempty }T\subseteq S.$$

Atkin–Lehner signs are constant on Galois orbits. Since every orbit in the family has degree $d$, passing from orbit sums to sums over conjugate eigenforms multiplies both $C_T$ and $N_d$ by $d$; the difficulty is therefore not the orbit weighting itself.

Local Plancherel heuristic

At a prime $\mathfrak q\nmid2D_F$ of conductor exponent one and trivial central character, the local representation is an unramified quadratic twist of the Steinberg representation. The two possibilities have opposite Atkin–Lehner signs and equal local Plancherel mass. Exact-new Plancherel equidistribution consequently predicts independent uniform signs in the ambient spectrum.

The exact rationality-degree condition is global and is not the eigenspace of a known trace-formula operator. Bounded-degree forms have density zero in the full spectrum, so ambient equidistribution does not determine their internal sign distribution. Excluding CM and base-change forms imposes two further global restrictions.

Why twisting does not prove the result

A quadratic twist can flip prescribed local Steinberg signs while preserving the Hecke field, but generally introduces auxiliary conductor. It may send levels bounded by $X$ to levels bounded by $cX$ rather than preserving the same cutoff, and it need not preserve the literal non-base-change family. A bijection between two differently scaled height ranges does not imply equality of same-cutoff densities without regular variation of the counting function. Narrow class number one also eliminates a general supply of nontrivial everywhere-unramified quadratic characters that would give a level-preserving involution.

Main obstacles

A proof requires fixed-degree correlation estimates

$$\sum_{O\in\mathcal H_{F,d,S}(X)} \prod_{\mathfrak q\in T}W_{\mathfrak q}(O) =o\bigl(\#\mathcal H_{F,d,S}(X)\bigr)$$

for every nonempty $T$, after removing CM and base-change orbits. No asymptotic count is known for the denominator itself, and no trace formula isolates exact Hecke-field degree. The analogous question is open even for classical weight-two newforms of a fixed rationality degree at varying prime level. A refined version of the conjecture may need to condition on the complete local representation type at primes dividing the discriminant of $F$ or at other systematically ramified places.

References

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