11 papers · 1 filter
The 2-adic Eigencurve is Proper
Kevin Buzzard, Frank Calegari
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…
Elliptic Curves of Odd Modular Degree
Frank Calegari, Matthew Emerton
The modular degree m_E of an elliptic curve E/Q is the minimal degree of any surjective morphism X_0(N) -> E, where N is the conductor of E. We give a necessarily set of criteria f…
Irrationality of certain p-adic periods for small p
Frank Calegari
Following Apery's proof of the irrationality of zeta(3), Beukers found an elegant reinterpretation of Apery's arguments using modular forms. We show how Beukers arguments can be ad…
Eisenstein Deformation Rings
Frank Calegari
We prove R = T theorems for certain reducible residual Galois representations. We answer in the positive a question of Gross and Lubin on whether certain Hecke algebras T are discr…
Mod p representations on elliptic curves
Frank Calegari
Modular Galois representations into GL_2(F_p) with cyclotomic determinant arise from elliptic curves for p = 2,3,5. We show (by constructing explicit examples) that such elliptic c…
On the Ramification of Hecke Algebras at Eisenstein Primes
Frank Calegari, Matthew Emerton
Using the modularity technique of Wiles, we study the Hecke algebra of weight 2 and prime level N localized at the Eisenstein primes. On the way, we recover some results of Mazur (…