### Twist-minimal trace formulas and applications

Heilbronn Number Theory Seminar

18th April 2018, 4:00 pm – 5:00 pm

Howard House, 4th Floor Seminar Room

One of the most well known examples of L-functions of degree 2 are L-functions of modular forms. Less well known, but equally important, are the L-funcitons of Maass forms. A Maass form is a function on a hyperbolic surface which is also an eigenfunction of the Laplace-Beltrami operator. Named after H. Maass, who discovered some examples in the 1940s, Maass forms remain largely mysterious.

Fortunately, there are concrete tools to study Maass forms: trace formulas, which relate the spectrum of the Laplace operator on a hyperbolic surface to its geometry. After Selberg introduced his famous trace formula in 1956, his ideas were generalised, and various trace formulas have been constructed and studied. However, there are few numerical results from trace formulas, the main obstacle being their complexity. Various types of trace formulas are investigated, constructed and used to understand automorphic representations and their L-functions from theoretical point of view, but most of them are not explicit enough to implement in computer code.

In this talk, we present a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0, and applications towards the Selberg eigenvalue conjecture and classification of 2-dimensional Artin representations of small conductor.

This is a joint work with Andrew Booker and Andreas Strömbergsson.

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