Level-raising of even representations of tetrahedral type and equidistribution of lines in the projective plane

Abstract: The distribution of auxiliary primes raising the level of even representations of tetrahedral type is studied. Under an equidistribution assumption, the density of primes raising the level of an even, $p$-adic representation is shown to be $$ \frac{p-1}{p}. $$ Data on auxiliary primes $v\leq 10^8$ raising the level of even $3$-adic representations of various conductors are presented. The data support equidistribution for $p=3$. In the process, we prove existence of even, surjective representations $$ \rho^{(\ell)}:\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \longrightarrow \operatorname{SL}(2,\mathbb{Z}_3) $$ ramified only at $\ell$ and at $3$ for $\ell =163$ and $\ell = 277$. The prime $\ell = 277$ falls outside the class of Shanks primes. Measured by conductor, these are the smallest known examples of totally real extensions of $\mathbb{Q}$ with Galois group $\operatorname{SL}(2, \mathbb{Z}_3)$.