Frazer Jarvis

On Galois representations associated to Hilbert modular forms.

In this paper, we prove that, to any Hilbert cuspidal eigenform, one may attach a compatible system of Galois representations. This result extends the analogous results of Deligne and Deligne–Serre for elliptic modular forms. The principal work on this conjecture was carried out by Carayol and Taylor, but their results left one case remaining, which we complete in this paper. We also investigate the compatibility of our results with the local Langlands correspondence, and prove that whenever the local component of the automorphic representation is not special, then the results coincide.

A Distribution Relation on Elliptic Curves

In this paper, we give an elementary proof of a curious identity of elliptic functions. It is very similar to a beautiful proof given by Coates of a different identity. The result was strongly motivated by Wildeshaus’s generalisation of Zagier’s conjecture.

Prime Ideals and Unique Factorisation

We have already studied unique factorisation in \(\mathbb {Z}\), and seen how it fails in certain rings of integers of number fields. We have also seen the suggestion that non-uniqueness of factorisation may be remedied by working with ideals. In order to show that this procedure will work generally, we will need to have some concept of what it means for an ideal to be prime.

Unique Factorisation in the Natural Numbers

We are so used to working with the natural numbers from infancy onwards that we take it for granted that natural numbers may be factorised uniquely into prime numbers. For example, \(360=2^{3}3^{2}5\) is the prime factorisation of 360. However, we should notice that there are already senses in which this factorisation is not really unique; we can write \(360=2\times 3\times 5\times 2\times 3\times 2\), or even \(360=(-2)\times 5\times 3\times (-3)\times 2\times 2\). Nevertheless, we can see that all these factorisations are “essentially the same”, in a way which we could make precise, and we will do so later.

Other Fields of Small Degree

The results of Chap. 6 give a fairly complete description of imaginary quadratic fields. But other fields have some different properties, and we will meet some of these for the first time in this chapter.

Imaginary Quadratic Fields

It won’t be a surpriseQuadratic field!imaginary|( that fields of low degree over \(\mathbb {Q}\) are going to be the easiest cases to understand.

Lattices and Geometrical Methods

In this chapter, we will prove two fundamental results in algebraic number theory: the finiteness of the class group, and Dirichlet’s Unit TheoremDirichlet, Peter Gustav Lejeune!Dirichlet’s unit theorem, which gives the structure of the group of units in the rings of integers of number fields.

Fields, Discriminants and Integral Bases

Discriminant Integral basisBy definition, every number field \(K\) is a finite extension of \(\mathbb {Q}\). In particular, if \(K\) has degree \(n\), then there must be elements \(\alpha _1,\ldots ,\alpha _n\in K\) such that every element of \(K\) can be written as a linear combination

ON A PAIRING BETWEEN SYMMETRIC POWER MODULES

x_1\alpha _1+x_2\alpha _2+\cdots +x_n\alpha _n