Let E be an elliptic curve over Q attached to a newform f of weight 2 on 00(N ), and let K be a real quadratic field in which all the primes dividing N are split. This paper relates the canonical R/Z-valued “circle pairing” on E(K ) defined by Mazur and Tate [MT1] to a period integral I ( f, K ) defined in terms of f and K . The resulting conjecture can be viewed as an analogue of the classical...

We discuss approaches to computing in the Shafarevich-Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of gen...

In this paper, we examine John Tate’s seminal work calculating functional equations for zeta functions over a number field k. Tate examines both ‘local’ properties of k, completed with respect to a given norm, and ‘global’ properties. The global theory examines the idele and adele groups of k as a way of encoding information from all of the completions of k into single structures, each with its...

We give a geometric proof of a “parity-switching” phenomenon that occurs when applying the local Langlands and Jacquet–Langlands correspondence to a self-dual supercuspidal representation ofGL(n) over a nonarchimedean local field. This turns out to reflect a duality property on the self-dual part of the `-adic étale cohomology of the Lubin–Tate tower.

We introduce method of approaching local class field theory from the perspective of Lubin-Tate formal groups. Our primarily aim is to demonstrate how to construct abelian extensions using these groups.

We provide a new and conceptually simplified construction of continuous homotopy fixed point spectra for Lubin-Tate spectra under the action of the extended Morava stabilizer group. Moreover, our new construction of a homotopy fixed point spectral sequence converging to the homotopy groups of the homotopy fixed points of Lubin-Tate spectra is isomorphic to an Adams spectral sequence converging ...

The existence of a good theory of Thom isomorphisms in some rational category of mixed Tate motives would permit a nice interpolation between ideas of Kontsevich on deformation quantization, and ideas of Connes and Kreimer on a Galois theory of renormalization, mediated by Deligne’s ideas on motivic Galois groups.