HALF-DERIVATIVES OF MODULAR FORMS AND INVARIANTS OF 3-MANIFOLDS

Don Zagier (MPI Bonn)

In the classical theory of modular forms, one associates to a modular (cusp) form

f(z) = ∑_{n>0} a(n) q^{n} (z ∈ ℌ = upper half-plane, q = e^{2πiz})

of integral weight k ≥ 2 on a subgroup Γ ⊂ SL(2, ℤ) its Eichler integral

f̃(z) = ∑_{n>0} \frac{a(n)}{n^{k−1}} q^{n} (z ∈ ℌ) ,

which satisfies (d/dz)^{k−1}f̃(z) = (const.) · f(z) and, as a consequence, also (cz + d)^{k−2}f̃(γ(z)) = f̃(z) + P_{γ}(z) for each γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Γ, with P_{γ}(z) a polynomial of degree ≤ k − 2 in z. The function f̃(z) has finite limiting values as z tends to a limit α ∈ ℚ, and in fact f̃(α + iε) is C^{∞} from the right at ε = 0 and has a terminating Taylor expansion ∑_{j=0}^{k−2} f̃_{j}(α)ε^{j} as ε ↘ 0 for all α ∈ ℚ, where the f_{j} : ℚ → ℂ are functions with interesting arithmetic properties (generalized modular symbols, Eichler-Shimura-Manin theory of periods).

The lecture discusses an analogue when f has weight 1/2, or more generally half-integral weight. For k = 1/2, the (k − 1)-fold integration defining the Eichler integral f̃ becomes the half-differentiation of the title. As an example, let η(z) = q^{1/24} ∏_{n=1}^{∞}(1 − q^{n}) be Dedekind’s eta-function, a modular form of weight 1/2 on the full modular group. Euler’s famous identity ∏_{n=1}^{∞}(1 − q^{n}) = ∑_{n∈ℤ}(−1)^{n}q^{(3n^{2}+n)/2} can be rewritten

η(z) = ∑_{n>0} χ_{12}(n) q^{n^{2}/24} , χ_{12}(n) = (\frac{12}{n}) = \{ \begin{matrix}1 & if n ≡ ±1 (mod 12), \\ −1 & if n ≡ ±5 (mod 12), \\ 0 & otherwise, \end{matrix}

so its (suitably normalized) “half-derivative” can be written

η̃(z) = ∑_{n>0} χ_{12}(n) n q^{n^{2}/24} = q^{1/24} (1 − 5q − 7q^{2} + 11q^{5} + 13q^{7} − ⋯) .

This function satisfies amusing identities like

\frac{η̃(z)}{η(z)} = 1 − 2 ∑_{n=1}^{∞} \frac{q^{n}}{1 − q^{n}} (1 + \frac{1}{(1 − q)(1 − q^{2}) ⋯ (1 − q^{n})}) ,

but the more interesting properties concern the behavior at or near rational points.

A couple of years ago, Kontsevich wrote down the sum

F (q) = ∑_{n=0}^{∞} (1 − q)(1 − q^{2}) ⋯ (1 − q^{n})

and observed that, although the series diverges both for |q| < 1 and |q| > 1, and also for most points on the unit circle, it terminates (and hence converges!) at all roots of unity, giving interesting values

F (1) = 1, F (−1) = 3, F (i) = 8 − 3i, …, F (q) ∈ ℤ[q]

whenever q is a root of unity. He also found experimentally an asymptotic formula

F (e^{2πi/k}) ∼ e^{−πi(k−3+1/k)/12} k^{3/2}

+ 1 + (−\frac{2πi}{k}) + \frac{3}{2!}(−\frac{2πi}{k})^{2} + \frac{19}{3!}(−\frac{2πi}{k})^{3} + ⋯

as k → ∞. One can also note that all (formal) derivatives of F make sense at all roots of unity, so that one obtains, for instance, expansions

F (q) = 1 + x + 2x^{2} + 5x^{3} + 15x^{4} + 53x^{5} + ⋯  (q = 1 − x, x → 0),
F (q) = 1 + \frac{t}{1!} + 3 \frac{t^{2}}{2!} + 19 \frac{t^{3}}{3!} + 207 \frac{t^{4}}{4!} + ⋯  (q = e^{−t}, t → 0),
q^{1/24}F (q) = 1 + 23 \frac{u}{1!} + 1681 \frac{u^{2}}{2!} + 257543 \frac{u^{3}}{3!} + ⋯  (q = e^{−24u}, u → 0)

as q → 1. Each of these expressions turns out to have an interesting interpretation (all three of which were first observed experimentally, but are now proved): the coefficients 1, 1, 2, 5, 15, … of the first expansion are equal to the numbers of “regularized linearized chord diagrams” as defined and computed by Stoimenow [3], and hence provide an upper bound, considerably better than the previously known ones, for the number of Vassiliev invariants of degree n; the coefficients 1, 1, 3, 19, … of the second expansion are the same as those occurring in Kontsevich’s empirical asymptotic expansion of F (e^{2πi/k}); and the coefficients 1, 23, 1681, … of the third expansion are equal to the “T-numbers” defined by Glaisher in 1898 by the generating function

\frac{sin 2X}{2 cos 3X} = \frac{X}{1!} + 23 \frac{X^{3}}{3!} + 1681 \frac{X^{5}}{5!} + ⋯,

and are also equal (up to a factor −2) to the values at negative even integers of the Dirichlet L-series associated to the character χ_{12} . The third relation is the one that connects with our “half-derivatives of modular forms”: these coefficient are the same as the coefficients in the power series expansion with respect to ε of the “Eichler integral” η̃(z) at z = iε for ε ↘ 0. More generally, one has the “identity”

F (q) = −\frac{1}{2} η̃(z) (q = e^{2πiz})

in which the two sides are never defined simultaneously but where the meaning is that the Taylor expansions (as power series in ε) of both sides for z = α + iε agree to all orders in ε for every rational number α. Another consequence is a modular property of the function η̃_{0} : ℚ → ℂ, which turns out to be a kind of modular form of weight 3/2 on ℚ modulo restrictions to ℚ of smooth functions on ℝ.

The same phenomenon of “functions of q with interesting behavior (only) at roots of unity” occurs in various quantum-theoretically inspired constructions in mathematics, and in particular in connection with quantum invariants of 3-dimensional manifolds. The second half of the lecture was devoted to this topic. Very briefly, the WRT (Witten-Reshitikhin-Turaev) invariant of a 3-manifold M is a “function” W_{M}(q) which was “defined” by Witten [4] in 1989 by a path integral over an infinite-dimensional space of connections on M to which no rigorous mathematical meaning can be assigned, and then defined combinatorially and rigorously by Reshitikhin and Turaev [2] in 1991 for q a root of unity. In recent work with Ruth Lawrence, it was shown that in a certain number of cases the function ℚ → ℂ defined by α ↦ W_{M}(e^{2πiα}) agrees with or is related to the Eichler integral in the sense described above of a modular form of half-integral weight. In particular, for the Poincaré homology sphere the values of W_{M}(q) are given by the Eichler integral of a theta series of weight 3/2 associated to an odd Dirichlet character of conductor 60. The details are given in [1].

References

[1] R. Lawrence and D. Zagier, Modular forms and quantum invariants of 3-manifolds. Asian J. Math. 3 (1999).

[2] N.Y. Reshetikhin and V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups. Invent. Math. 103 (1991), 547–597.

[3] A. Stoimenow,Enumeration of chord diagrams and an upper bound for Vassiliev invariants. J. of Knot Theory and its Ramifications 7 (1998), 93–114.

[4] E. Witten, Quantum Field Theory and the Jones Polynomial. Commun. Math. Phys. 121 (1989), 351–399.

[5] D. Zagier, Vassiliev invariants and a strange identity related to the Dedekind eta-function. Preprint, MPI 1999.