# Elliptic functions and modular invariance

### Time and place

Thursdays 10-12 in room DK117, starting September 19th

### Schedule and summaries of presentations

**Thursday, September 19, 10-12 in room DK117:** Introduction (Konstantin Izyurov)

- Weierstrass' algebraic addition theorem

**Thursday, September 26:** NO STUDY GROUP. There's a seminar by Benjamin Doyon instead.

**Thursday, October 3, 10-12 in room DK117:** Elliptic functions, modular functions and modular forms (Christian)

- Some general discussion about what modular functions and forms are and what kind of fields do they appear in.
- Latter half of Chapter 1 of Apostol's book - constructing a modular function from the Weierstrass elliptic function.

**Thursday, October 10, 10-12 in room DK117:** Klein's modular function and its Fourier series (Kay)

**Thursday, October 17, 10-12 in room DK117:** More on Klein's modular function (Miika)

- generators of the modular group, fundamental domain for the modular group action on H, zeroes and poles at the conical singularities, modular functions as rational functions of J

**Thursday, October 24, 10-12 in room DK117:** Eta function (Kalle)

- modular forms of fractional weight and multiplier system, modular properties of the eta-function (by Poisson summation, for the easy way see Apostol's book instead), some combinatorics of partitions (subexponential growth of the number of partitions, Euler's pentagonal number theorem)

**Thursday, October 31, 10-12 in room DK117:** Zeta-function regularization of the determinant of Laplacian (Ali)

**Thursday, November 7, 10-12 in room DK117:** Theta functions (Kostya).

- definitions, transformation properties, zeros.
- relation to elliptic functions, heat equation and Poisson kernel in annuli.
- More-or-less-modularity of theta constants. Uniformization of the universal cover of thrice punctured Riemann sphere.

**Thursday, November 14, 10-12 in room DK117: **Leftover topics

- Quadratic reciprocity via modular property of theta constants (Kostya)
- Terms of the Rademacher series for number of partitions, see notes attached to the previous talk (Kalle)

**Thursday, November 21, 10-12 in room DK117:** Dimensions of the spaces of modular forms of a given weight (Christian)

**Thursday, December 5, 10-12 in room DK117:** Two 16-dimensional drums that sound the same (Eve)

- Lattices, their theta series, modularity of theta series of even unimodular lattices, two isospectral non-isometric flat tori in dimension 16.

**Thursday, December 12:** NO STUDY GROUP. There's a noon-to-noon stochastic sauna seminar.

**Thursday, December 19:** Asymptotics of determinant of the Laplacian on discrete tori (Kostya): Notes based on Complexity and heights of tori by Chinta, Jorgensen and Karlsson.

### Description

The idea is to cover some basics about elliptic functions and modular forms, peppered with as many nice applications as possible. Some such applications, in no particular order, could be for example:

- Weierstrass' addition theorem
- The Eynard-Kristjansen solution of O(n) model on a random lattice [Eynard&Kristjansen]
- Some transfer matrix solution of a lattice model from Baxter's book [Baxter]
- The Kac–Wakimoto Conjecture, idea explained in [Zagier, p.31]
- Drums Whose Shape One Cannot Hear, idea explained in [Zagier, p.36]
- Moments of Periodic Functions, idea explained in [Zagier, p.50]
- The Irrationality of ζ(3), idea explained in [Zagier, p.64]
- An Example Coming from Percolation Theory, idea explained in [Zagier, p.66]
- Zeta-function regularization of the determinant of the Laplacian on torus, idea explained in [Gawedzki, p.10], briefly discussed in [Di Francesco et al, Chapter 10.2]
- Modularity of characters of unitary c<1 representations of the Virasoro algebra, [Rocha-Caridi], briefly discussed in [Di Francesco et al, Chapters 10.5-10.7]

### Literature

One good-looking basic book about elliptic and modular functions is:

- T. Apostol: "Modular functions and Dirichlet series in number theory". Springer, 1976.

Everyone is invited to find nice applications of the theory and present them to the others. Many nice applications can be found in the following two:

- D. Zagier: "Elliptic Modular Forms and Their Applications". 2008 (available online).
- P. Di Francesco, P. Mathieu & D. Sénéchal: "Conformal Field Theory", Chapter 10. Springer, 1997.

The following cound contain more about the determinant of the Laplacian, which is a nice application:

- K. Gawedzki: "Lectures on Conformal Field Theory". IAS 1995.
- J. Dubédat: "Dimers and analytic torsion". arXiv:1110.2808.

More direct applications to statistical mechanics are for example:

- B. Eynard & C. Kristjansen: "Exact Solution of the O(n) Model on a Random Lattice" and a sequel to it. arXiv:hep-th/9506193 and arXiv:hep-th/9512052
- R. J. Baxter: "Exactly solved models in statistical mechanics". Academic Press, 1982.