# Masterclass: Mahler measures and special values of L functions

## University of Copenhagen, 27-31 August 2018

This Masterclass is meant to illustrate to young researchers the latest developments in the study of special values of L-functions and its connection to Mahler’s measure, polylogarithms, hypergeometric functions and K-theory.

## Prerequisites and references:

The courses will require only a basic background in algebraic number theory and algebraic geometry and will be as self-contained as possible.

For a detailed list of references and preliminary reading we kindly refer to this PDF file.

## Lecture series:

François Brunault (École normale supérieure de Lyon)

To be announced

## Abstracts:

### François Brunault: Mahler measures, regulators and modular forms

The first lecture will be devoted to the construction of Beilinson's regulator for $K_2$ of algebraic curves and their function fields, with an emphasis on explicit formulas. We will give the definition of tame symbols and their links with residues. Following Boyd, Deninger and Rodriguez Villegas, we will explain how to use Jensen's formula to express the Mahler measure of a two-variable polynomial in terms of regulators for the corresponding curve.

More generally, we will explain how to define Deligne cohomology of complex algebraic varieties and discuss some cases where the differential forms can be made explicit. We will also give a brief survey of the properties of motivic cohomology and the definition of $L$-functions of varieties over number fields. Armed with this machinery, we will explain Deninger's method and its variants to relate multivariate Mahler measures to $L$-values in the framework of Beilinson's conjectures, generalizing the case of curves.

We will next explain how modular methods can be used to tackle particular cases of Boyd's conjectures on Mahler measures. Specifically, we will describe Beilinson's construction of the Eisenstein symbols (an algebraic version of Eisenstein series) in the motivic cohomology of elliptic curves. This construction turns out to be very useful in the case of modular curves and their associated universal elliptic curves, which we will also discuss. Using this, we will define the regulator integrals and give their main properties.

After a survey of known results and conjectures on multivariate Mahler measures, due to many people, we will explain the method initiated by Rogers and Zudilin to compute these regulator integrals in terms of $L$-values, first in the simple case of the Boyd-Deninger polynomial $x+1/x+y+1/y+1$, corresponding to an elliptic curve of conductor $15$, and then in greater generality for higher weight modular forms. As Neururer will explain in his talk, this leads to new proofs and generates new identities for Mahler measures of universal elliptic curves. We will explain the expected properties of the modular forms appearing in the formulas for regulator integrals. Finally, we will focus on some methods to find "nice" equations for powers of universal elliptic curves.

### Wadim Zudilin: Many variables of Mahler measures

The course starts with some basics in algebraic number theory and a historical account of the Mahler measure of univariate polynomials/algebraic numbers.
The review includes origin of Lehmer's problem about the minimal Mahler measure, Mahler's own application of the measure in the theory of transcendental numbers, and simple relations between the logarithmic Mahler measure, regulators and $L$-values through Dirichlet's class number formula.
We then proceed to some partial resolutions of Lehmer's problem, due to Smyth, Dobrowolski and others, and to a related problem of Odlyzko and Serre about discriminants of polynomials.

Though the multidimensional generalization given by Mahler in the 1960s was targeted at its usefulness in transcendental numbers, potential applications of the multivariate Mahler measure in Lehmer's problem and a peculiar range of this generalized characteristic have made it a subject of independent study.

In our exposition of the (logarithmic) Mahler measure for polynomials in several variables, we highlight analytical aspects of the story focusing mainly on hypergeometric expressions of the measure and related differential equations.
This part goes in parallel with the material in lectures of François Brunault, where the logarithmic Mahler measure is linked with the regulators and $L$-values of algebraic varieties defined by the zero loci of corresponding multivariate polynomials.

We conclude the course with relating the Mahler measure in three or more variables to planar random walk.

### Michalis Neururer: Mahler measures of elliptic surfaces

The (logarithmic) Mahler measure of a polynomial $P$ in $n$ variables is defined as the mean of $\log \lvert P \rvert$ restricted to the real n-torus.
In 1997 Deninger noticed a remarkable connections between the Mahler measure of a polynomial and period integrals of the associated algebraic variety. When P defines certain elliptic curves these period integrals are related to L-values by Beilinson's conjectures.

In the following years many conjectured relations between Mahler measures of polynomials defining elliptic curves and the L-values (at $s=2$) of these curves were proved.
In my talk I will give an overview of this topic and discuss my current joint work with François Brunault, where we study the Mahler measure of a polynomial defining a K3-surface and relate it to the L-value (at $s=3$) of the surface.

## Financial support:

The conference is funded by the Faculty of Science of the University of Copenhagen.
We will only be able to provide funding for the speakers.

## Venue(s):

The Masterclass will take place at the Department of Mathematical Sciences of the University of Copenhagen. The room for all the classes and the talks will be announced in due time.

There will be a beer tasting in the lunch room (04.4.19) on the fourth floor at 18.30 on Monday. There will also be a conference dinner at 19.00 on Thursday.

## Registration:

We kindly ask you to register for the conference by filling in this form.

The deadline for registering is July 27, 2018.

## Participants:

 First name Last name Affiliation Xiaohua Ai Max-Planck-Institut für Mathematik, Bonn Waheed Ali Amur Jitendra Bajpai Georg-August-Universität Göttingen Sofiane Bouarroudj New York University Abu Dhabi Daniele Dona Georg-August-Universität Göttingen Xuejun Guo Nanjing University Mounir Hajli School of Mathematical Sciences, Shanghai Jiao Tong University, China Kenneth Jensen University of Copenhagen Qingzhong Ji Nanjing University Odile Lecacheux UPMC Sorbonne Université Hang Liu Shaanxi Normal University Roderick McCrorie University of St Andrews Mahya Mehrabdollahei Sorbonne Université, Paris, France Katharina Müller Universität Göttingen Hourong Qin Nanjing University Sumaia Saad Eddin JKU university Linz Deividas Sabonis TU Munich&Uni Copenhagen Ritika Sharma Harish-Chandra Research Institute Allahabad, India Matteo Verzobio University of Pisa, Italy Weijia Wang UMPA, ENS Lyon, France

## How to get to Copenhagen

Copenhagen is served by a big airport with many international flights to all the major European cities. To look up flights to Copenhagen you can use the airport’s route map or SkyScanner. Alternatively you can consider to come here by train (see the route planner of Deutsche Bahn) or by bus (for example with Flixbus).

## Accommodation

We kindly ask the participants to arrange their own accommodation. We will recommend some options for accommodation, whose locations can be found on the venue(s) map.

Some recommended options for hotels are:

Since all these options might be a bit expensive for younger participants, we include also a recommended list of hostels:

We recommend to check also AirBnB for private accommodation.

## Public transportation

Tickets and passes for public transportation can be bought at the Copenhagen Airport and every train or metro station. You can find the DSB office on your right hand side as soon as you come out of the arrival area of the airport. DSB has an agreement with 7-Eleven, so many of their shops double as selling points for public transportation. A journey planner in English is available. More information on the "find us" webpage.

Organised by Fabien Pazuki and Riccardo Pengo.