The geometric level-raising conjecture for the eigencurve

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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 x∈Ex\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 Z⊂E\mathcal{Z}\subset\mathcal{E} is a generically principal-series at ll component and there exists x∈Zx\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 Z′⊂E\mathcal{Z}'\subset\mathcal{E} such that x∈Z′x\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.

References

Primary source

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

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