Root-number equidistribution for genuine rational Bianchi newforms

Abstract

We conjecture that the two global root numbers occur with equal limiting frequency among genuine non-CM rational Bianchi newforms of parallel weight two and trivial central character over a fixed imaginary quadratic field, ordered by level norm. We describe the finite-level bias visible in current data and explain the obstruction created by constant-sign quadratic-twist families.

The family and the conjecture

Retain the notation $\mathcal R_K(X)$ and $\mathcal L_K(X)$ for rational non-CM Bianchi newforms and their twisted-base-change subfamily. Define the genuine family by

$$\mathcal G_K(X)=\mathcal R_K(X)\setminus\mathcal L_K(X).$$

For $F\in\mathcal G_K(X)$, let $w(F)\in\{\pm1\}$ be the sign in the functional equation of its completed standard $L$-function.

Conjecture 1. For every imaginary quadratic field $K$, if $\#\mathcal G_K(X)\to\infty$, then for each $\varepsilon\in\{\pm1\}$,

$$\lim_{X\to\infty} \frac{\#\{F\in\mathcal G_K(X):w(F)=\varepsilon\}} {\#\mathcal G_K(X)}=\frac12.$$

Equivalently, if

$$A_K(X)=\#\mathcal G_K(X), \qquad S_K(X)=\sum_{F\in\mathcal G_K(X)}w(F),$$

then Conjecture 1 is the assertion

$$S_K(X)=o\bigl(A_K(X)\bigr).$$

Numerical evidence

The available data contain $215090$ genuine non-CM rational orbits, of which $98391$ have sign $+1$ and $116699$ have sign $-1$. The pooled proportion is not the conjectured statistic because the level coverage differs strongly from one field to another.

For the deepest fixed-field datasets, the terminal positive proportions are:

$\mathop{\mathrm{Disc}}(K)$	$-3$	$-4$	$-7$	$-8$	$-11$	$-19$	$-23$	$-31$
$\Pr(w=+1)$	.4470	.4459	.4514	.4573	.4729	.5005	.4712	.4725

Positive root-number proportions at the largest available fixed-field cutoffs.

The deepest fields show an initial bias toward sign $-1$ that weakens as the norm cutoff grows. For example, the field of discriminant $-3$ moves from positive proportion $0.4280$ at norm $10000$ to $0.4470$ at norm $149968$. Both signs occur in every meaningfully populated fixed-field sample.

Heuristic motivation

The global root number is a product of local signs. Once CM forms and all base-change mechanisms are removed, no visible global symmetry forces one sign. This suggests cancellation in a sufficiently large fixed-field level family, in analogy with classical sign-equidistribution results.

The rationality and genuineness restrictions are global and very thin. A trace formula on the full newspace gives a weighted sum

$$\sum_F [\mathbb Q_F:\mathbb Q]w(F),$$

not the unweighted sum over rational genuine orbits required here. There is no known projector imposing simultaneously $\mathbb Q_F=\mathbb Q$, non-CM status and exclusion of every twisted-base-change orbit.

The quadratic-twist obstruction

There are rational genuine level-one forms over $K=\mathbb Q(\sqrt{-643})$. Let $\pi$ be one such form. For a quadratic Hecke character $\chi/K$, twisting preserves parallel weight $2$, rational Hecke eigenvalues, trivial central character, non-CM status and genuineness. Since $\pi$ is unramified at every finite place, the local twisting formula and the product formula give

$$w(\pi\otimes\chi)=w(\pi)$$

for every quadratic $\chi$. Non-CM status rules out nontrivial quadratic self-twists, so this produces infinitely many distinct genuine rational forms with the same root number.

This does not disprove Conjecture 1: the constant-sign twist family may have density zero inside $\mathcal G_K(X)$. It does show that equidistribution cannot be proved by a naive sign-reversing quadratic-twist pairing.

Relation with known results

Classical trace-formula results prove root-number equidistribution in full newspaces, and recent automorphic results treat self-dual families in other aspects. These theorems do not isolate the rational genuine Bianchi locus. Existing structural work describes the genuine/non-genuine decomposition and provides examples of rational genuine forms, but no signed counting theorem in the level aspect is known.

Open difficulties

For each fixed $K$, one needs both a sufficiently regular lower bound for $A_K(X)$ and the signed estimate

$$\sum_{F\in\mathcal G_K(X)}w(F)=o\bigl(A_K(X)\bigr).$$

It is unknown whether the explicit constant-sign quadratic-twist families are negligible, have positive density, or are balanced by other twist classes. A proof would therefore require a counting theory for genuine rational Bianchi twist classes that is substantially finer than current dimension formulae.

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