In this paper, we prove many cases of the anticyclotomic main conjecture for general CM fields with p-ordinary CM type.

Abstract. Let p ≥ 5 be a prime. If an irreducible component of the spectrum of the ‘big’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its mod p Galois representation contains an open subgroup of SL2(Fp[[T]]) for the canonical “weight ” variable T. This fact appears to be deep, as it is almost equivalent to the vanishing of the µ-invariant of the Kubota–Leopoldt p-adic L-function and the anticyclotomic Katz p-adic L-function and another key ingredientof the proof is the anticylotomic main conjecture proven by Rubin/Mazur–Tilouine. Fix a prime p ≥ 5, field embeddings C i∞ ← ↪ Q ip ↩ → Qp ⊂ Cp and a positive integer N prime to p. We sometimes identify Cp and C by a fixed field isomorphism compatible with the above embeddings. Consider the space of modular forms Mk+1(Γ0(Np r+1), χ) with (p ∤ N, r ≥ 0) (including Eisenstein series) and cusp forms Sk+1(Γ0(Np r+1), χ). Let the ring Z[χ] ⊂ C and Zp[χ] ⊂ Q p be generated by the values χ over Z and Zp, respectively. The Hecke algebra H = Hk+1(Γ0(Np r+1), χ; Z[χ]) over Z[χ] is

We prove the modularity of minimally ramified ordinary residually reducible p-adic Galois representations of an imaginary quadratic field F under certain assumptions. We first exhibit conditions under which the residual representation is unique up to isomorphism. Then we prove the existence of deformations arising from cuspforms on GL2(AF) via the Galois representations constructed by Taylor et al. We establish a sufficient condition (in terms of the non-existence of certain field extensions which in many cases can be reduced to a condition on an L-value) for the universal deformation ring to be a discrete valuation ring and in that case we prove an R = T theorem. We also study reducible deformations and show that no minimal characteristic 0 reducible deformation exists.

Abstract. Let p ≥ 5 be a prime. If an irreduciblecomponent of the spectrum of the ‘big ’ ordinary Hecke algebra does not have complex multiplication, under mild assumptions, we prove that the image of its Galois representation contains, up to finite error, a principal congruence subgroup Γ(L) of SL2(Zp[[T]]) for a principal ideal (L) ̸ = 0 of Zp[[T]] for the canonical “weight ” variable T. If nontrivial (i.e., L ̸ = 1), the power series L is proven to be a factor of the Kubota-Leopoldt p-adic L-function or of the square of the anticyclotomic Katz p-adic L-function (or a power of of (1 + T) pm − 1)). Fix a prime p ≥ 3 throughout the paper, field embeddings C i∞ ← ↪ Q ip ↩ → Qp ⊂ Cp and a positive integer N prime to p. Let Λ = Zp[[T]], and write Q for its quotient field. Consider the space of modular forms Mk+1(Γ0(Np r+1), χ) with (p ∤ N, r ≥ 0) (with Eisenstein series) and cusp forms Sk+1(Γ0(Np r+1), χ). Let Z[χ] ⊂ Q and Zp[χ] ⊂ Q p be the rings generated by the values χ over Z and Zp, respectively. The Hecke algebra H = Hk+1(Γ0(Np r+1), χ; Z[χ]) over Z[χ] is

We review some recent results on the arithmetic of the theta correspondence for certain symplectic-orthogonal dual pairs and some applications to periods and congruences of modular forms. We also propose an integral version of a conjecture on Petersson inner products of modular forms on quaternion algebras over totally real fields.