TY - JOUR
T1 - Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity
AU - Uttenthal, Peter Vang
PY - 2024/5
Y1 - 2024/5
N2 - This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 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.
AB - This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p=2 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.
KW - Arithmetic statistics
KW - Deformation theory of Galois representations
KW - Density theorems
KW - Even Galois representations
KW - Galois cohomology
KW - Global class field theory
KW - Global reciprocity
KW - Selmer groups
KW - Wiles' formula
UR - http://www.scopus.com/inward/record.url?scp=85182370050&partnerID=8YFLogxK
U2 - 10.1016/j.jnt.2023.11.010
DO - 10.1016/j.jnt.2023.11.010
M3 - Journal article
SN - 0022-314X
VL - 258
SP - 212
EP - 268
JO - Journal of Number Theory
JF - Journal of Number Theory
ER -