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Let f be a newform of weight k ≥ 3 with Fourier coefficients in a number field K. We show that the universal deformation ring of the mod λ Galois representation associated to f is unobstructed, and thus isomorphic to a power series ring in three variables over the Witt vectors, for all but finitely many primes λ of K. We give an explicit bound on such λ for(More)
This study assessed the variability of six goniometric measurements commonly used in the assessment of children with cerebral palsy (CP). Three experienced paediatric physiotherapists recorded three consecutive measurements of six joint ranges from 12 children with spastic CP. A fourth measurement was recorded 1 week later. The order of measurement with(More)
ρ̄f,λ : GQ,S∪{`} → GL2 kλ over the residue field kλ of K at λ; here GQ,S∪{`} is the Galois group of the maximal extension of Q unramified outside the set S of places dividing N∞ and the characteristic ` of kλ. The representation ρ̄f,λ is absolutely irreducible for almost all primes λ; we write Red(f) for the set of λ such that ρ̄f,λ is not absolutely(More)
We expect this conjecture to be false (for sufficiently large d) when E does have complex multiplication. Nevertheless, our main result is that Conjecture 1.1 at least holds on average. For A,B > 0, let SA,B denote the set of elliptic curves with Weierstrass equations y = x + ax + b with a, b ∈ Z, |a| ≤ A and |b| ≤ B. For an elliptic curve E and x > 0, let(More)
Let ρ̄ : GQ → GL2(k) be an absolutely irreducible modular Galois representation over a finite field k of characteristic p. Assume further that ρ̄ is p-ordinary and p-distinguished in the sense that the restriction of ρ̄ to a decomposition group at p is reducible and non-scalar. The Hida family H(ρ̄) of ρ̄ is the set of all p-ordinary p-stabilized newforms f(More)
In fact, we suspect that much more is true: we conjecture that this relative density does not change after restriction to any set of primes defined by a Cebatorevstyle Frobenius condition; that is, we expect that the sets (0.1) yield sets of primes of positive density which are quite different from those sets determined by Galois theoretic conditions. See(More)
The concept of quantum monodromy is introduced to give insight into the energy levels of systems with cylindrically symmetrical potential energy barriers. The K structure of bending progressions of bent molecules and the pendular states of dipolar molecules in strong electric ® elds are taken as molecular examples. Results are given for a two-dimensional(More)
Fix a squarefree integer N and let f be a newform of weight 2 for Γ0(N); we assume that f does not have complex multiplication. It was shown in [14] and [15] that for a set of primes l of density 1 the naive deformation theory of the mod l Galois representation associated to f is unobstructed (in the sense that the universal deformation ring is a power(More)