Fix a prime p > 2. Let ρ : Gal(Q/Q) → GL2(I) be the Galois representation coming from a non-CM irreducible component I of Hida's p-ordinary Hecke algebra. Assume the residual representationρ is absolutely irreducible. Under a minor technical condition we identify a subring I0 of I containing Zp [[T ]] such that the image of ρ is large with respect to I0. That is, Im ρ contains ker(SL2(I0) → SL2(I0/a)) for some non-zero I0-ideal a. This paper builds on recent work of Hida who showed that the image of such a Galois representation is large with respect to Zp [[T ]]. Our result is an I-adic analogue of the description of the image of the Galois representation attached to a non-CM classical modular form obtained by Ribet and Momose in the 1980s.
It is known that infinitely many number fields and function fields of any degree m have class number divisible by a given integer n. However, significantly less is known about the indivisibility of class numbers of such fields. While it's known that there exist infinitely many quadratic number fields with class number indivisible by a given prime, the fields are not constructed explicitly, and nothing appears to be known for higher degree extensions. In [32], Pacelli and Rosen explicitly constructed an infinite class of function fields of any degree m, 3 ∤ m, over F q (T ) with class number indivisible by 3, generalizing a result of Ichimura for quadratic extensions. Here we generalize that result, constructing, for an arbitrary prime ℓ, and positive integer m > 1, infinitely many function fields of degree m over the rational function field, with class number indivisible by ℓ.
Abstract. Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K) ⊗ Qp, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q) ⊗ Qp. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We conclude by illustrating these algorithms in a collection of examples.
Shimura and Taniyama proved that if A is a potentially CM abelian variety over a number field F with CM by a field K linearly disjoint from F, then there is an algebraic Hecke character λA of F K such that L(A/F, s) = L(λA, s). We consider a certain converse to their result. Namely, let A be a potentially CM abelian variety appearing as a factor of the Jacobian of a curve of the form y e = γx f + δ. Fix positive integers a and n such that n/2 < a ≤ n. Under mild conditions on e, f, γ, δ, we construct a Chow motive M , defined over F = Q(γ, δ), such that L(M/F, s) and L(λ a A λ n−a A , s) have the same Euler factors outside finitely many primes.
For an odd prime p, we study the image of a continuous 2-dimensional (pseudo)representation ρ of a profinite group with coefficients in a local pro-p domain A. Under mild conditions, Bellaïche has proved that the image of ρ contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We prove that the ring B can be slightly enlarged and then described in terms of the conjugate self-twists of ρ, symmetries that naturally constrain its image; hence this new B is optimal. We use this result to recover, and in some cases improve, the known large-image results for Galois representations arising from elliptic and Hilbert modular forms due to Serre, Ribet and Momose, and Nekovář, and p-adic Hida or Coleman families of elliptic modular forms due to Hida, Lang, and Conti-Iovita-Tilouine.
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