Related papers: The Eigencurve is Proper

We prove that the Coleman-Mazur eigencurve is proper (over the weight space) at a large class of points.

We prove that the Coleman-Mazur eigencurve is proper (over weight space) at integral weights in the center of weight space.

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…

Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of…

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 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…

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…

In this paper we show that the critical Eisenstein series of weight 2, E_{2}^{crit_{p}}, is smooth in the eigencurve C(l), where l is a prime. We also show that E_{2}^{crit_{p},ord_{l}} is smooth in the full eigencurve C^{full}(l) and…

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.

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…

Mazur's isogeny theorem states that if $p$ is a prime for which there exists an elliptic curve $E / \mathbb{Q}$ that admits a rational isogeny of degree $p$, then $p \in \{2,3,5,7,11,13,17,19,37,43,67,163 \}$. This result is one of the…

Let p be a prime, C the p-adic Eigencurve (with tame level 1) and Z the blow-up of the Fredholm hypersurface of the U_p - operator at the special points. We show that for p = 2, 3, 5 and 7, the natural map C -> Z is a rigid-analytic…

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 three-dimensional torus $(\mathbb{R}/\mathbb{Z})^3$, all the eigenvalues having an eigenfunction which satisfies the Courant nodal domains theorem with equality…

We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero…

We study an eigenvalue problem for prescribed $\sigma_k$-curvature equations of star-shaped, $k$-convex, closed hypersurfaces. We establish the existence of a unique eigenvalue and its associated hypersurface, which is also unique, provided…

The main result of this paper is an instance of the conjecture made by Gouvea and Mazur (Math. Res. Lett., 1995) which asserts that for certain values of r the space of r-overconvergent p-adic modular forms of tame level N and weight k…

In this paper, we proved the normal scalar curvature conjecture and the Bottcher-Wenzel conjecture.

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric K\"ahler metrics with non-negative scalar curvature. In…