On flat even deformation rings

In the presence of a nontrivial dual Selmer group, certain global even deformation rings are shown to be finite and flat over $\mathbb{Z}_p$. Previously, flatness was only known in established cases of Langlands reciprocity in the odd parity. By techniques from global class field theory, explicit examples of even representations are computed to which the results apply. For even representations $\overlineρ$ in an explicit family, it is observed that if Leopoldt’s conjecture is true for a certain number field attached to $\overlineρ$, then the global even deformation ring is flat at the minimal level.

💡 Research Summary

The paper addresses a long‑standing gap in the deformation theory of Galois representations: while the flatness of global deformation rings over ℤₚ has been well understood for odd two‑dimensional residual representations (those for which complex conjugation has determinant −1), the analogous question for even representations (determinant +1) remained largely open, especially when both the Selmer group and its dual are non‑trivial.

The author adopts the Galois cohomological method, which reformulates deformation problems purely in terms of Selmer conditions and their duals, without relying on automorphic or geometric input. A Selmer condition N is a collection of local subspaces N_v ⊂ H¹(G_v, Ad⁰ ρ) for each place v in a finite set S containing p and ∞. Its dual N^⊥ is defined via local Tate pairings, and the global Selmer and dual Selmer groups are the kernels of the natural restriction maps.

A central notion introduced is a balanced global setting: the Selmer and dual Selmer groups have the same ℱₚ‑dimension. The paper focuses on the rank‑one case, i.e. both groups are one‑dimensional. In this situation, the Weierstrass preparation theorem implies that any global deformation ring R_N is isomorphic to a quotient
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