Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in even residual characteristic obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL(2): At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families.Comment: 44 page

Existence of pp-adic representations and flatness of even deformation rings in balanced global settings of rank one

The main goal of this paper is to prove existence of $p$-adic lifts of even Galois representations in balanced global settings with dual Selmer groups of rank one, and to construct examples of even residual representations in characteristic $p=3$ for which the theorem yields existence of $3$-adic lifts. In the proof of the main theorem, we show that if the global deformation rings are not flat over $\mathbb{Z}_p$ at the minimal level, then after allowing ramification at one auxiliary prime, the new parts of the global rings enjoy the flatness property. The proof is independent of the parity of the representations. Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity, while it is new for even deformation rings in balanced global settings of rank one. As a corollary, we show that by choosing the auxiliary prime appropriately, we can control the rank mod $p$ of the new part of the flat ring. The examples of $3$-adic representations and flat, even deformation rings are built by applying techniques from global class field theory to a family of totally real fields studied by D. Shanks. In the process of constructing the even residual representations, we show that there exist cuspidal automorphic representations $\pi$ and $\pi'$ of $\operatorname{GL}(2,\mathbb{A}_{\mathbb{Q}})$ attached to classical Maass wave forms of level $\ell=20887$ whose mod 3 reductions are distinct. The proof applies a theorem of R. P. Langlands from his theory of base change

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

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 $\operatorname{SL}(2, \mathbb{Z}_3)$.Comment: 22 pages, 4 tables, 1 figure. arXiv admin note: text overlap with arXiv:2309.0187