Growth of Selmer groups of Hilbert modular forms over ring class fields

Growth of Selmer Groups of Hilbert Modular Forms over Ring Class Fields

On the Vanishing of Selmer Groups for Elliptic Curves over Ring Class Fields

Let E/Q be an elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N . Assume that the number of primes dividing N and inert in K is odd, and let Hc be the ring class field of K of conductor c prime to ND with Galois group Gc over K. Fix a complex character χ of Gc. Our main result is that if LK(E,χ, 1) 6= 0 then Selp(...

Deformations of Hilbert Modular Galois Representations and Adjoint Selmer Groups

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to a Hilbert modular form under mild assumptions on the residual image. In particular, we do not assume the residual image satisfies the Taylor–Wiles hypothesis. Using this, we deduce that the localization and completion of the universal deformation ring for the r...

Large Selmer groups over number fields

Let p be a prime number and M a quadratic number field, M , Q( √ p) if p ≡ 1 mod 4. We will prove that for any positive integer d there exists a Galois extension F/Q with Galois group D2p and an elliptic curve E/Q such that F contains M and the p-Selmer group of E/F has size at least pd.

Kolyvagin’s descent and Mordell-Weil groups over ring class fields

whereX0(N) is the complete curve overQ which classifies pairs of elliptic curves related by a cyclic N -isogeny. The curve E is equipped with the collection of Heegner points defined over ring class fields of suitable imaginary quadratic fields. More precisely, let K be an imaginary quadratic field in which all rational primes dividing N are split and let O be the order of K of conductor c prim...