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# Number Theory

## Can every even integer greater than 2 can be expressed as the sum of two primes? Which integers can be the area of a right-angled triangle with integer length sides? Are there infinitely many pairs of prime numbers of the form (p, p+2)? Are there infinitely many real quadratic number fields with unique factorisation?

These are all important open questions in number theory. In its most basic sense, number theory is the study of the properties of the integers and their building blocks, the prime numbers. Number theory is an ancient mathematical discipline that remains a thriving area of modern research, using tools from algebra, analysis and geometry.

### RESEARCH

#### The geometry and arithmetic of Shimura varieties (Chris Daw)

Shimura varieties are playgrounds for many kinds of mathematics. Although arising as geometric objects, they possess rich algebraic and arithmetic structure.

Goro Shimura introduced many examples of Shimura varieties as he sought to generalise the fact that every algebraic integer whose Galois group is abelian can be expressed as a sum of roots of unity with rational coefficients, otherwise known as the Kronecker-Weber theorem. Since then, they have played a key role in the Langlands programme, providing a precious source of Galois and automorphic representations.

A question going back to Robert Coleman was, up to isomorphism, how many algebraic curves of genus g have the property that their Jacobian (an abelian variety) has complex multiplication (CM)?  This is really a question about the moduli space of principally polarised abelian varieties of dimension g , which is an example of a Shimura variety. It led to the so-called André-Oort conjecture, which says that subvarieties of Shimura varieties having lots of CM points must be Shimura varieties themselves.

There is now a whole world of conjectures on the geometry and arithmetic of Shimura varieties. The underlying theme is that of unlikely intersections, which refers to the phenomenon of when a subvariety intersects many Shimura subvarieties of small dimension.

#### Analytic number theory (Titus Hilberdink)

Number theory contains a wealth of fascinating and easily stated problems, both solved and unsolved, which can be tackled using many different mathematical techniques; to mention just a few, the Prime Number Theorem, Fermat's Last Theorem, the ABC conjecture, and perhaps the greatest open problem of all – the Riemann Hypothesis. Our interests lies in using methods from complex analysis and functional analysis to study number theoretical objects.

The theory of Beurling primes generalizes the usual prime numbers where only a multiplicative structure is assumed. The associated zeta function can be studied using complex analysis, giving new insights into the actual primes.

Functional Analysis plays a role when viewing Dirichlet convolution as an operator on suitable spaces. This leads to the notion of multiplicative Toeplitz operators and their matrix representations which have a rich arithmetical structure. Some of this links to work of Dr J Virtanen.

#### Rational points on algebraic varieties (Rachel Newton)

Given a polynomial equation with rational coefficients, we may ask:

1. Does it have any rational solutions?

2. If so, how many? Finitely many? Can we find all of them? Infinitely many? Do the solutions have any interesting structure? Can we generate new solutions from old ones? Can we find some finite collection from which we can generate all the others?

Given a family of equations, we can ask what proportion of them have rational solutions.

The study of rational points on algebraic varieties encompasses the arithmetic of elliptic curves and abelian varieties, local-global principles such as the Hasse principle, and obstructions to these local-global principles. We can use geometric machinery alongside algebraic techniques, such as class field theory, as well as analytic methods to study rational solutions in a family of equations.

### People

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