The geometric level-raising conjecture for the eigencurve

Let E\mathcal{E} be the cuspidal eigencurve associated with a modular two-dimensional residual representation, and let (ρx,l,Nx)(\rho_{x,l},N_x) be the Weil–Deligne representation attached to xEx\in\mathcal{E}. A component is generically principal series or generically special at ll according to the corresponding generic local representation type.

Geometric level-raising conjecture. If ZE\mathcal{Z}\subset\mathcal{E} is a generically principal-series at ll component and there exists xZx\in\mathcal{Z} such that π(ρx,l,Nx)\pi(\rho_{x,l},N_x) is one dimensional, then there exists a generically special irreducible component ZE\mathcal{Z}'\subset\mathcal{E} such that xZx\in\mathcal{Z}'.

This is presented as a geometric analogue of Ribet's level-raising theorem and, together with the preceding statement, as equivalent to the main eigencurve intersection conjecture. The supplied text gives no resolution.

Sources & referencesView supporting material

Primary source

Alexander G. M. Paulin, “Failure of the Local to Global Principle in the Eigencurve”, arXiv:1001.2051 (2010).

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