# Oxford Mathematics Research probes the inner secrets of L-functions

Oxford Mathematician Tom Oliver talks about his research in to the rich mine of mathematical information that are L-functions.

"I am interested in the analytic properties of L-functions, which are supposed to encode deep arithmetic information. The best known L-function is the Riemann zeta function, which is intimately connected to the theory of prime numbers. More general L-functions are attached to algebraic varieties and automorphic forms - the former being solutions to systems of polynomials, and the latter being analytic functions on certain symmetric spaces of arithmetic interest. An elaborate web of conjectures, known as the Langlands programme, relates the two.

The L-functions of automorphic forms have well-documented analytic behaviour. My recent research attempts to answer the converse question - if we know an L-function has (weakened versions of) such analytic properties, is there an underlying automorphic form? The aim of the game is to make the assumptions as weak as possible. In doing so, we open the door to various applications. For example, the so-called 'grand simplicity hypothesis' suggests that the non-real zeros of automorphic L-functions are simple and linearly independent (over the field of rational numbers). It is exceptionally difficult to prove anything like this in wide generality, so I am interested in special cases.

In order to get a handle on what is achievable, one has to understand ways in which L-functions can be more or less complicated than one another. A common way to quantify the complexity of an L-function is a numerical invariant called the degree. Intuitively, this is the number of gamma functions appearing in the functional equation. For example the Riemann zeta function has degree 1, and the degree grows with the degree of the base field, the dimension of the algebraic variety, the rank of the algebraic group, etc. The higher the degree, the more 'trivial' zeros an L-function has. These are the zeros on the real line cancelling the poles of the gamma function.

In a recent collobaration with Michalis Neururer (TU Darmstadt), we studied the simplest proxy for independence of zeros imaginable, which is still unresolved even for small degrees. Namely, when dividing two L-functions, one would expect that the quotient has infinitely many poles (zeros of the denominator which are not zeros of the numerator). This is easily seen to be the case when the degree of the numerator is less than that of the denominator, as the denominator has more trivial zeros. These are the negative degree quotients. More interesting are quotients of positive degree, where the poles correspond to non-trivial zeros.

In 2013, Andrew Booker proved that degree 0 and degree 1 quotients have infinitely many poles. Building upon Booker's ideas, our work focused on the case that the difference in degree is 2. Assuming that the quotient has only finitely many poles, we show that the quotient looks enough like an automorphic L-function (in fact, the L-function of a Maass form), to apply a suitably general converse theorem. This gives rise to a factorisation of the numerator, contradicting orthogonality results for automorphic L-functions. The argument involves the construction of special symmetries of the hyperbolic plane."