We study the arithmetic of Eisenstein cohomology classes (in the sense of G. Harder) for symmetric spaces associated to GL2 over imaginary quadratic fields. We prove in many cases a lower bound on their denominator in terms of a special L-value of a Hecke character providing evidence for a conjecture of Harder that the denominator is given by this L-value. We also prove under some additional as...

We prove the vanishing of the geometric Bloch–Kato Selmer group for the adjoint representation of a Galois representation associated to regular algebraic polarized cuspidal automorphic representations under an assumption on the residual image. Using this, we deduce that the localization and completion of a certain universal deformation ring for the residual representation at the characteristic ...

Manjul Bhargava and I have recently proved a result on the average order of the 2-Selmer groups of the Jacobians of hyperelliptic curves of a fixed genus n ≥ 1 over Q, with a rational Weierstrass point [2, Thm 1]. A surprising fact which emerges is that the average order of this finite group is equal to 3, independent of the genus n. This gives us a uniform upper bound of 3 2 on the average ran...

We provide a relation between the $\mu $-invariants of dual plus and minus Selmer groups for supersingular elliptic curves when we ascend from cyclotomic ${\mathbb Z}_p$-extension to Z}_p^2$-extension over an imaginary quadratic field.

We develop a theory of ‘non-abelian higher special elements’ in the non-commutative exterior powers Galois cohomology p-adic representations. explore their relation to organising matrices and thus module structure Selmer modules. In concrete applications, we relate our general formulation refined conjectures Birch Swinnerton-Dyer type Tate–Shafarevich groups abelian varieties.

We present a number of papers on topics in mathematics and theoretical computer science. Topics include: a problem relating to the ABC Conjecture, the ranks of 2-Selmer groups of twists of an elliptic curve, the Goldbach problem for primes in specified Chebotarev classes, explicit models for Deligne-Lusztig curves, constructions for small designs, noise sensitivity bounds for polynomials thresh...

Let C : y = f(x) be a hyperelliptic curve defined over Q. Let K be a number field and suppose f factors over K as a product of irreducible polynomials f = f1f2 . . . fr. We shall define a “Selmer set” corresponding to this factorization with the property that if it is empty then C(Q) = ∅. We shall demonstrate the effectiveness of our new method by solving the generalized Fermat equation with si...

We derive the equations necessary to perform a two-descent on the Jacobians of curves of genus two with rational Weierstrass points. We compute the Mordell-Weil rank of the Jacobian of some genus two curves defined over the rationals, and discuss the practicality of using this method. Introduction Let C he a curve of genus two defined over a number field K, and J its Jacobian variety. The Morde...