Related papers: On properness of the eigencurve
We prove that the Coleman-Mazur eigencurve is proper (over weight space) at integral weights in the center of weight space.
We prove that the Coleman-Mazur eigencurve is proper over the weight space for any prime p and tame level N.
For p=2 and tame level N=1 we prove that the map from the (Coleman-Mazur) Eigencurve to weight space satisfies the valuative criterion of properness. More informally, we show that the Eigencurve has no "holes"; given a punctured disc of…
We give a new proof of the properness of the Coleman-Mazur eigencurve. The question of whether the eigencurve satisfies the valuative criterion for properness was first asked by Coleman and Mazur in 1998 and settled by Diao and Liu in 2016…
We prove that the eigencurve associated to a definite quaternion algebra over $\QQ$ satisfies the following properties, as conjectured by Coleman--Mazur and Buzzard--Kilford: (a) over the boundary annuli of weight space, the eigencurve is a…
We show that the p-adic Eigencurve is smooth at classical weight one points which are regular at p and give a precise criterion for etaleness over the weight space at those points. Our approach uses deformations of Galois representations.
We prove the Halo conjecture on the geometry of the eigencurve over the boundary of the weight space, predicted by Coleman-Mazur and Buzzard-Kilford.
We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.
We show that any space with a positive upper curvature bound has in a small neighborhood of any point a closely related metric with a negative upper curvature bound.
In this paper, we determine, in the case of the Laplacian on the flat two-dimensional torus (R/Z) 2 , all the eigenvalues having an eigenfunction which satisfies Courant's theorem with equality (Courant-sharp situation). Following the…
It is proved that a parameterized curve in a metric space $X$ is absolutely continuous if and only if its composition with any Lipschitz function on $X$ is absolutely continuous.
This paper generalizes work of Buzzard and Kilford to the case $p=3$, giving an explicit bound for the overconvergence of the quotient $E_\kappa / V(E_\kappa)$ and using this bound to prove that the eigencurve is a union of countably many…
We study the group of automorphisms of the affine plane preserving some given curve, over any field. The group is proven to be algebraic, except in the case where the curve is a bunch of parallel lines. Moreover, a classification of the…
In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.
These are notes based on four lectures given at the Heidelberg spring school on non-archimedean geometry and eigenvarieties. None of the contents are original work. Our goal is to explain the construction of eigenvarieties in various…
A new generalization of the classical separate algebraicity theorem is suggested and proved.
A short proof of the Mazur-Ulam theorem concerning isometries of real normed spaces.
In this paper, as a result of a theorem of Serre on congruence properties, a complete solution is given for an open question (see the text) presented recently by Kim, Koo and Park. Some further questions and results on similar types of…
We prove that entire conformal curves $\mathbb{R}^n \rightarrow \mathbb{R}^m$ fall into two classes: either the curve is affine or the average energy in a ball is strictly increasing for large radii and diverges to infinity. This rigidity…
Let p be a prime number and C be the p-adic tame level 1 eigencurve introduced by Coleman-Mazur. We prove that C is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight…