Root Numbers of Binary-Tetrahedral Artin Representations

Abstract

We consider faithful irreducible two-dimensional Artin representations of $G_{\mathbf Q}$ with image $\mathrm{SL}_2(\mathbf F_3)$, trivial determinant, and Frobenius–Schur indicator $-1$. Ordered by Artin conductor, we conjecture that this family is infinite and that its global root numbers are equidistributed. We also record a proof of infinitude by quadratic twisting and explain why this construction does not itself produce sign cancellation.

The family and the conjecture

Let $q(\rho)$ and $w(\rho)\in\{\pm1\}$ denote the Artin conductor and global root number of a finite-image complex representation $\rho$ of $G_{\mathbf Q}$. For $X\ge1$, set

$$\begin{aligned} \mathcal T(X)=\bigl\{[\rho]:{}&\rho:G_{\mathbf Q}\to\mathrm{GL}_2(\mathbf C) \text{ is continuous, faithful, and irreducible},\\ &\operatorname{im}\rho\simeq\mathrm{SL}_2(\mathbf F_3),\quad \operatorname{FS}(\rho)=-1,\\ &\det\rho=1,\quad q(\rho)\le X\bigr\}. \end{aligned}$$

where representations are counted up to complex isomorphism. No restriction is imposed on the projective $A_4$-field or on the set of ramified primes.

Conjecture 1. The cardinality of $\mathcal T(X)$ tends to infinity, and

$$\lim_{X\to\infty}\frac{1}{\#\mathcal T(X)} \sum_{[\rho]\in\mathcal T(X)}w(\rho)=0.$$

Binary-tetrahedral representations form the first non-dihedral, projectively exceptional symplectic family in dimension two. Tetrahedral automorphy makes their root numbers analogous to Atkin–Lehner signs, while variation of the projective $A_4$-field and of the local lifting data provides a plausible source of cancellation.

Numerical evidence

The recorded LMFDB data give the following cumulative sign counts.

$X$	$w=+1$	$w=-1$
$10^5$	4	4
$2.5\cdot10^5$	12	18
$5\cdot10^5$	34	42
$10^6$	49	60
$1.7\cdot10^6$	56	73

Cumulative root-number counts. At the last cutoff, $129$ of the $133$ recorded orbits have known sign.

Both signs occur repeatedly. The full target table contains $278$ rows and shows no determinant variation or hidden duplication in the inspected fields.

An infinitude result

The infinitude clause of Conjecture 1 follows from a quadratic-twist construction.

Proposition 2. There are infinitely many isomorphism classes of representations satisfying all the defining conditions of $\mathcal T(X)$.

Proof. Caputo and Vinatier construct a tame Galois extension $N_0/\mathbf Q$ with group $G\simeq\mathrm{SL}_2(\mathbf F_3)$ and a faithful irreducible symplectic representation $\rho_0$ of degree two and root number $-1$. Let $q_0=q(\rho_0)$.

For every prime $\ell\equiv1\pmod4$ with $\ell\nmid q_0$, let $\chi_\ell$ be the primitive quadratic character of conductor $\ell$ and put $\rho_\ell=\rho_0\otimes\chi_\ell$. Since $G^{\mathrm{ab}}\simeq C_3$, the extension $N_0$ is linearly disjoint from $\mathbf Q(\sqrt\ell)$. The joint image of $(\rho_0,\chi_\ell)$ is therefore $G\times\{\pm1\}$, and the map $(g,e)\mapsto eg$ has image $G$ and kernel $\{(I,1),(-I,-1)\}$. Hence $\operatorname{im}\rho_\ell\simeq G$.

Scalar quadratic twisting preserves irreducibility, the alternating form, and the determinant. At $\ell$, inertia acts by the scalar $-I$, so the inertia-fixed space is zero and the local conductor exponent is two; at all other primes the conductor is unchanged. Thus

$$q(\rho_\ell)=q_0\ell^2.$$

Distinct primes give distinct conductors. Dirichlet’s theorem consequently yields

$$\#\mathcal T(X)\ge \#\left\{\ell\equiv1\pmod4:\ \ell\nmid q_0, \ \ell\le\sqrt{X/q_0}\right\}\longrightarrow\infty.$$

 ◻

This family does not prove the sign assertion. The local epsilon-factor computation used by Shankar–Södergren–Templier gives $w(\rho_0\otimes\chi_\ell)=w(\rho_0)=-1$ for these coprime twists. Thus the conjectural cancellation must occur between different projective $A_4$-fields, different reduced-Schur lifting classes, or different wild local types, rather than inside the twists of a single representation.

Relation to existing results and remaining problem

Rubinstein-Salzedo conjectures equidistribution of a reduced Schur lifting invariant in a restricted tame, totally real family of $A_4$-fields. Dalal and Gerbelli-Gauthier prove root-number equidistribution in a varying-weight automorphic family, but their hypotheses do not cover a fixed finite-image conductor aspect. Even the required conductor-ordered count of the relevant $A_4$-fields with prescribed central lift and local conditions is not currently available.

Writing

$$S(X)=\sum_{[\rho]\in\mathcal T(X)}w(\rho),$$

the unresolved assertion is $S(X)=o(\#\mathcal T(X))$. A proof would require sufficiently uniform counting, ordered by the two-dimensional Artin conductor, for liftable $A_4$-fields with prescribed local conditions and prescribed lifting or root-number invariant. The finite computation is consistent with the conjecture but does not establish this signed asymptotic.

References

• Luca Caputo and Stéphane Vinatier, Galois module structure of the square root of the inverse different in even degree tame extensions of number fields (2016). https://doi.org/10.1016/j.jalgebra.2016.06.035

• Simon Rubinstein-Salzedo, Invariants for $A_4$ fields and the Cohen–Lenstra heuristics (2014). https://arxiv.org/abs/1210.2773

• Arul Shankar, Anders Södergren, and Nicolas Templier, Sato–Tate equidistribution of certain families of Artin L-functions (2019). https://doi.org/10.1017/fms.2019.18

• Rahul Dalal and Mathilde Gerbelli-Gauthier, Root Number Equidistribution for Self-Dual Automorphic Representations on $\mathrm{GL}_N$ (2025). https://arxiv.org/abs/2410.01976

• Robert P. Langlands, Base Change for GL(2) (1980). https://publications.ias.edu/book/export/html/64

• Daniel Loughran and Tim Santens Paterson, Lower bounds for counting $A_4$-quartic fields (2025). https://arxiv.org/abs/2510.05248

• Philippe Michel and Akshay Venkatesh, On the dimension of the space of cusp forms associated to 2-dimensional complex Galois representations (2002). https://doi.org/10.1155/S1073792802206121