Rankin–Cohen bracket

In mathematics, the Rankin–Cohen bracket of two modular forms is another modular form, generalizing the product of two modular forms. Rankin (1956, 1957) gave some general conditions for polynomials in derivatives of modular forms to be modular forms, and Cohen (1975) found the explicit examples of such polynomials that give Rankin–Cohen brackets. They were named by Zagier (1994), who introduced Rankin–Cohen algebras as an abstract setting for Rankin–Cohen brackets.

Definition

If $f(\tau )$ and $g(\tau )$ are modular forms of weight k and h respectively then their nth Rankin–Cohen bracket [f,g]_n is given by

$[f,g]_{n}={\frac {1}{(2\pi i)^{n}}}\sum _{r+s=n}(-1)^{r}{\binom {k+n-1}{s}}{\binom {h+n-1}{r}}{\frac {\mathrm {d} ^{r}f}{\mathrm {d} \tau ^{r}}}{\frac {\mathrm {d} ^{s}g}{\mathrm {d} \tau ^{s}}}\ .$

It is a modular form of weight k + h + 2n. Note that the factor of $(2\pi i)^{n}$ is included so that the q-expansion coefficients of $[f,g]_{n}$ are rational if those of $f$ and $g$ are. $d^{r}f/d\tau ^{r}$ and $d^{s}g/d\tau ^{s}$ are the standard derivatives, as opposed to the derivative with respect to the square of the nome which is sometimes also used.

Representation theory

The mysterious formula for the Rankin–Cohen bracket can be explained in terms of representation theory. Modular forms can be regarded as lowest weight vectors for discrete series representations of SL₂(R) in a space of functions on SL₂(R)/SL₂(Z). The tensor product of two lowest weight representations corresponding to modular forms f and g splits as a direct sum of lowest weight representations indexed by non-negative integers n, and a short calculation shows that the corresponding lowest weight vectors are the Rankin–Cohen brackets [f,g]_n.

Rings of modular forms

The zero-th Rankin–Cohen bracket is the Lie bracket when considering a ring of modular forms as a Lie algebra.

References

• Cohen, Henri (1975), "Sums involving the values at negative integers of L-functions of quadratic characters", Math. Ann., 217 (3): 271–285, doi:10.1007/BF01436180, MR 0382192, Zbl 0311.10030
• Rankin, R. A. (1956), "The construction of automorphic forms from the derivatives of a given form", J. Indian Math. Soc., New Series, 20: 103–116, MR 0082563, Zbl 0072.08601
• Rankin, R. A. (1957), "The construction of automorphic forms from the derivatives of given forms", Michigan Math. J., 4: 181–186, doi:10.1307/mmj/1028989013, MR 0092870
• Zagier, Don (1994), "Modular forms and differential operators", Proc. Indian Acad. Sci. Math. Sci., K. G. Ramanathan memorial issue, 104 (1): 57–75, doi:10.1007/BF02830874, MR 1280058, Zbl 0806.11022