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Mahler measures and special values of L functions
Provider: Faculty of Science

Activity no.: 5597-18-07-31 
Enrollment deadline: 27/08/2018
PlaceDepartment of Mathematical Sciences
Date and time27.08.2018, at: 09:00 - 31.08.2018, at: 16:00
Regular seats30
ECTS credits2.50
Contact personNina Weisse    E-mail address: weisse@math.ku.dk
Enrolment Handling/Course OrganiserFabien Pazuki    E-mail address: fpazuki@math.ku.dk
Written languageEnglish
Teaching languageEnglish
Course workload
Course workload categoryHours
Class Seminar10.00
Exercises25.00
Course hours20.00
Lectures10.00
Evaluation / reporting1.00
Sum66.00
Content
We wish to attract at the University of Copenhagen a variety of international young researchers working in this area with different backgrounds to introduce them to the very fascinating links between Mahler measures and special values of L-functions.

The aim of the course will be to introduce PhD students and young postdocs to Mahler measures, L-functions and their connection with modular and Siegel units, hypergeometric functions, polylogarithms and K-theory.

In order to do so we have contacted two of the main experts in the area: François Brunault and Wadim Zudilin. Brunault has proved several cases of Boyd's conjectures (see [9], [10], [11], [12]) by giving an explicit description of the algebraic cycle defined by Deninger and using an explicit version of Beilinson's theorem that he proved in his PhD thesis (see [9]). He has proved cases of Boyd's conjectures also for curves of genus greater than one (see [10]) and for elliptic surfaces (see [12]). Zudilin has also proved several cases of Boyd's conjectures by relating the Mahler measure of a family of polynomials with hypergeometric functions, in the cases of elliptic curves (see [15], [17] and [18]) and curves of higher genus (see [2] and [3]). He has also proved a substantial explicit expression for the regulator of two modular units (see [19]) which was generalized to Siegel units by Brunault (see [11]). Both of them are internationally known researchers and lecturers.

Formal requirements
We expect the participants to be familiar with basic notions of algebraic number theory, algebraic geometry and complex analysis.

We kindly ask the participants to read the following papers and notes:
-- [1], [6] and [8] to have a good introduction to the subject;
-- [7], which is the paper in which Boyd asked many of the conjectures are taken;
-- [13] and [16] for the link with Beilinson's conjectures in K-theory.

Learning outcome
We expect the participants to be actively involved in the lectures, and to participate in the various exercise sessions.
By the end of the masterclass the participants should be able to understand the main results and open conjectures relating Mahler measures and special values of L-functions.

Literature
Suggested references:

[1] M. Bertin, "Mahler's measure: From Number Theory to Geometry". In J. McKee & C. Smyth (Eds.), Number Theory and Polynomials (London Mathematical Society Lecture Note Series, pp. 20-32). Cambridge: Cambridge University Press (2008).
https://doi.org/10.1017/CBO9780511721274.004

[2] M. J. Bertin and W. Zudilin, "On the Mahler measure of a family of genus 2 curves", Math. Zeitschrift 283:3 (2016), 1185–1193.
https://doi.org/10.1007/s00209-016-1637-6

[3] M. J. Bertin and W. Zudilin, "On the Mahler measure of hyperelliptic families". Ann. Math. Québec 41:1 (2017), 199–211.
https://doi.org/10.1007/s40316-016-0068-4

[4] S. Bloch and K Kato, "L-functions and Tamagawa numbers of motives". The Grothendieck Festschrift. Birkhäuser Boston, 2007. 333-400.
https://doi.org/10.1007/978-0-8176-4574-8_9

[5] H. Bornhorn, "Mahler measures, K-theory and values of L-functions." arXiv preprint arXiv:1503.06069 (2015).
https://arxiv.org/abs/1503.06069

[6] D. W. Boyd, D. Lind, F. Rodriguez-Villegas and C. Deninger, “The many aspects of Mahler’s measure”, Banff, 2003.
https://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf

[7] D. W. Boyd, "Mahler's measure and special values of L-functions." Experimental Mathematics 7.1 (1998): 37-82.
https://doi.org/10.1080/10586458.1998.10504357

[8] F. Brunault, "Zagier's conjectures on special values of L-functions". Rivista di Matematica della Università di Parma 7 (2004), n°3*, pp. 165-176.
http://rivista.math.unipr.it/fulltext/2004-3s/pdf/08.pdf

[9] F. Brunault, "Version explicite du théorème de Beilinson pour la courbe modulaire X_1(N)". Comptes Rendus Mathématique, Académie des Sciences, Paris 343 (2006), pp. 505-510.
https://doi.org/10.1016/j.crma.2006.09.014

[10] F. Brunault, "On the Mahler measure associated to X_1(13)". arXiv preprint arXiv:1503.04631 (2015).
https://arxiv.org/abs/1503.04631

[11] F. Brunault, "Regulators of Siegel units and applications". Journal of Number Theory 163 (2016), pp. 542-569.
https://doi.org/10.1016/j.jnt.2015.12.019

[12] F. Brunault and M. Neururer, "Mahler measures of elliptic modular surfaces". arXiv preprint arXiv:1710.04602 (2017).
https://arxiv.org/abs/1710.04602

[13] C. Deninger, "Deligne Periods of Mixed Motives, K-Theory and the Entropy of Certain {{Z} n-Actions". Journal of the American Mathematical Society 10.2 (1997): 259-281.
https://doi.org/10.1090/S0894-0347-97-00228-2

[14] C. Deninger and A.J. Scholl, "The Beilinson conjectures". London Math. Soc. Lecture Notes 153 (1991): 173-209.
https://doi.org/10.1017/CBO9780511526053.007

[15] M. Lalín, D. Samart and W. Zudilin, "Further explorations of Boyd's conjectures and a conductor 21 elliptic curve", J. London Math. Soc. (2) 93:2 (2016), 341–360.
https://doi.org/10.1112/jlms/jdv073

[16] F. Rodriguez-Villegas. "Modular Mahler measures. I". In "Topics in number theory (University Park, PA, 1997)", volume 467 of Math. Appl., pages 17–48. Kluwer Acad. Publ., Dordrecht, 1999.
https://doi.org/10.1007/978-1-4613-0305-3_2

[17] M. Rogers and W. Zudilin, "From L-series of elliptic curves to Mahler measures". Compositio Math. 148:2 (2012), 385–414.
https://doi.org/10.1112/S0010437X11007342

[18] M. Rogers and W. Zudilin, "On the Mahler measure of 1+X+1/X+Y+1/Y". Intern. Math. Research Notices 2014:9 (2014), 2305–2326.
https://doi.org/10.1093/imrn/rns285

[19] W. Zudilin, "Regulator of modular units and Mahler measures". Math. Proc. Cambridge Philos. Soc. 156:2 (2014), 313–326.
https://doi.org/10.1017/S0305004113000765

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