# Cologne Young Researchers in Number Theory Program 2015

### Instructors: Dr. Larry Rolen Second Instructor: Michael Griffin

Participants:

Alexandru Ciolan (Rheinische Friedrich-Wilhelms-Universität Bonn)

Benjamin Engel (Universität zu Köln)

Johannes Girsch (Universität Wien)

Minjoo Jang (Yonsei University)

Jacob Manaker (Swarthmore College)

Robert Neiss (Universität zu Köln)

Basic Information:

Thanks to the generous support of the University of Cologne and the DFG, I am delighted to announce the Cologne Young Researchers in Number Theory Program in the spring of 2015 here in beautiful Cologne, Germany. The aim of this program is to offer a competitive, intensive 6-week program for 6-8 advanced students, especially those interested in pursuing a research career in mathematics. The program will begin with one week of lectures on modular forms and related topics, after which students will jump into original research projects. Students will work on teams of 2-3 people with the goal of publishing, or at least writing up a comprehensive summary of their findings.

Students will be reimbursed for travel expenses (up to a reasonable amount) and provided with free housing in an historic apartment building centrally located in Ehrenfeld, Cologne.

Eligibility:

Students from Europe or abroad are welcome to apply. Previous experience in number theory is not required, however students should have a solid understanding of basic algebra and analysis, and applications will be competitive. Undergraduate and Master's students are welcome to apply, but students who are undertaking or have previously started a Ph.D. program in mathematics are not eligible.

Application Process:

To apply, please send a single PDF containing a personal statement, a list of all mathematics courses taken together with grades in these courses, a CV, and arrange to have two letters of recommendation sent on your behalf to larryrolen@gmail.com. The deadline is October 15, 2014.

Project Areas:

We are very excited about the projects offered in this year's program, and projects will be related to modular forms, harmonic Maass forms, or other similar number theory topics. A few sample topics include the following (but please note that projects are subject to change):

1. Singular moduli: In this project, students would begin by learning the beautiful theory of complex multiplication, which states that certain special values of the j-function are algebraic integers which generate Hilbert class fields of imaginary quadratic fields. For example, this amazing fact is related to Hermite's classical observation that $e^{\pi\sqrt{163}}$ is "very nearly" an integer (type it into a computer and see!) Students would then learn about recent connections between singular moduli of non-holomorphic modular functions to harmonic Maass forms and Poincaré series. In particular, students would learn about the pathbreaking paper of Bruinier-Ono which gives a finite (algebraic!) formula for the partition function. Students would then have several possible directions of study. For example, they could investigate topics such as: Gross-Zagier-type formulas, irreducibility properties of non-holomorphic Hilbert class polynomials, and relations between singular moduli and classical results on combinatorial generating functions.

Those interested may find the following papers useful:

J. Bruinier and K. Ono, Algebraic Formulas for the Coefficients of Half-Integral Weight Harmonic Weak Maass Forms, Advances in Mathematics, 246 (2013), pages 198-219.

V. Dose, N. Green, M. Griffin, T. Mao, L. Rolen, and J. Willis, Class Polynomials for a Distinguished Non-Holomorphic Modular Function, to appear in Proc. of the Amer. Math. Soc.

M. Griffin and L. Rolen, Integrality Properties of Symmetric Functions in Singular Moduli, submitted.

E. Larson and L. Rolen, Integrality Properties of the CM-Values of Certain Weak Maass Forms, to appear in Forum Math.

D. Zagier, Traces of Singular Moduli, "Motives, Polylogarithms and Hodge Theory" (Eds. F. Bogomolov, L. Katzarkov), Lecture Series 3, International Press, Somerville (2002), 209-244.

2. Structure of Hecke algebras: In recent work by Nicolas and Serre, much of the explicit structure and effective bounds on "nilpotency degrees" for Hecke algebras modulo 2 have been determined. In particular, they give very explicit bounds which state that when one applies enough Hecke operators, one kills modular forms modulo 2. Students could use these findings to study several possible aspects. The first, and simplest, would be to investigate applications to explicit congruences of modular forms of these results. Furthermore, there is much computational and theoretical work to be done yet in certain higher level cases in which local nilpotency is known, but not explicitly. Along seperate lines, studies of the explicit structure of Hecke algebras could yield further computational evidence for higher-level Weil converse conjectures, which state that q-series transforming as eigenfunctions of the Hecke operators and the Fricke involutions must be modular forms.

Those interested may find the following papers useful:

J. Conrey and D. Farmer, An Extension of Hecke's Converse Theorem, Internat. Math. Res. Notices 1995, no. 9, 445-463.

J-L Nicolas and J-P Serre, Formes Modulaires Modulo 2: l'Ordre de Nilpotence des Opérateurs de Hecke, C. R. Math. Acad. Sci. Paris 350 (2012), no. 7-8, 343-348.

3. Infinite families of sums of squares and related identities: A celebrated theorem of Legendre states that every positive integer can be expressed as the sum of at most 4 squares of natural numbers. This can be proven using the theory of modular forms, which Jacobi used to provide explicit formulas for the number of representations of an integer as a sum of 4 or 8 squares. In a monumental work, Milne presented infinite families of identities generalizing these and other classical identities in the theory of modular forms. This work depends on very special continued fraction expansions of quotients of Jacobi elliptic functions, the theory of Hankel determinants, and other classical theories. In this project, students would explore previously understudied continued fraction expansions of more exotic elliptic functions to try to provide new classes of identities along these lines.

Those interested may find the following paper useful:

S. Milne, Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic functions, Continued Fractions, and Schur Functions, Ramanujan J. 6 (2002), no. 1, 7-149.

4. Period polynomials: Period polynomials are classical objects with encode important analytic data attached to modular forms and provide the basis for the famous Eichler-Shimura theory, which provides isomorphisms of spaces of modular forms to spaces of polynomials with certain relations. Recent work of Pasol and Popa generalizes this theory in a comprehensive way to higher level. This work affords numerous questions which need addressing. For example, students could study conjectures regarding the location of the zeros of these polynomials, as well as the relation between period ratios and nontrivial elements in certain geometrical groups. Furthermore, preliminary calculations using Magma show surprising data which begs an explanation.

5. Those interested may find the following papers useful:

J. Conrey, D. Farmer, and Ö. Imamoglu, The Nontrivial Zeros of Period Polynomials of Modular Forms Lie on the Unit Circle, Int. Math. Res. Not. IMRN 2013, no. 20, 4758-4771.

N. Dummigan, Period Ratios of Modular Forms. Math. Ann. 318 (2000), no. 3, 621-636.

A. El-Guindy and W. Raji, Unimodularity of Roots of Period Polynomials of Hecke Eigenforms, Accepted in Bulletin of London Math Society.

V. Pasol and A. Popa, Modular Forms and Period Polynomials, Proc. Lond. Math. Soc. (3) 107 (2013), no. 4, 713-743.

6. Quantum modular forms: The field of quantum modular forms was only recently founded in 2010 by Zagier in his eponymous survey paper. Unlike classical modular forms, these objects "live" on the rationals, i.e., on the cusps of the modular group. Moreover, they satisfy a "near-modularity property", where the "error to modularity" is no longer zero, but in some sense a nice function. Although this definition is (intentionally) vague, an explosion of recent papers have presented numerous strange and beautiful examples connected to very important objects, for example knot invariants, ranks and cranks of partitions, mock theta functions, partial theta functions, and Kac-Wakimoto characters which arise in Lie theory. In this project, students would have several possible avenues of exploration. One of the missing pieces of the current field of quantum modular forms is a cohesive theory which gives an arithmetic or analytic theorem for a large, general class of quantum modular forms (indeed, the field is so young, that in many cases, finding examples of them is a worthwhile theorem in itself). Students could investigate a comprehensive theory of congruences for these objects, inspired by recent work of Andrews and Sellers. Students could also explore unstudied aspects of quantum modular forms, such as distributional questions on their values, and students would also be able to search for new examples of these strange objects.

Those interested may find the following papers useful:

G. Andrews and J. Sellers, Congruences for the Fishburn Numbers, submitted, arxiv:1401.5345.

K. Bringmann, T. Creutzig, and L. Rolen, Negative Index Jacobi Forms and Quantum Modular Forms, submitted, arxiv: 1401.7189.

A. Folsom, K. Ono, and R. Rhoades, Mock Theta Functions and Quantum Modular Forms, Forum of Mathematics, Pi, 1 (2013), e2, 27 pages.

L. Rolen and R. Schneider, On the Construction of a Natural Vector-Valued Quantum Modular Form, Arxiv der Mathematik (July 2013).

D. Zagier, Quantum Modular Forms, Quanta of Maths : Conference in honor of Alain Connes, Clay Mathematics Proceedings 11, AMS and Clay Mathematics Institute 2010, 659-675.

7. Meromorphic modular forms: Modular forms with poles on the upper-half plane have a more complicated theory and have received relatively little attention in the literature. However, meromorphic modular forms have shown up in numerous recent papers and have brought these objects more to the forefront. Thus, it is reasonable to ask about their arithmetic properties. Using recent work of Berndt and Bialek inspired by certain formulas of Ramanujan, several important numerical properties have been observed by the organizer. The project would be to prove these arithmetic properties for a class of meromorphic modular forms. Students would learn about the Circle Method and other analytic techniques.

Those interested may find the following papers useful:

C. Alfes, M. Griffin, K. Ono, and L. Rolen, Weierstrass Mock Modular Forms and Elliptic Curves, submitted, arxiv: 1406.0443.

B. Berndt and P. Bialek, On the Power Series Coefficients of Certain Quotients of Eisenstein Series, Trans. of the Amer. Math. Soc., Volume 357, Number 11, Pages 4379-4412.