MATH Seminar

Title: Modular linear differential equations
Seminar: Algebra
Speaker: Kiyokazu Nagatomo of Osaka University
Contact: David Zureick-Brown,
Date: 2019-09-03 at 4:00PM
Venue: MSC W303
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The most naive definition of \textit{modular linear differential equations} (MLDEs) would be linear differential equations whose space of solutions are invariant under the weight $k$ slash action of $\Gamma_1=SL_2(\mathbb{Z})$, for some $k$. Then under an analytic condition for coefficients functions and the Wronskians of a~basis of the space of solutions of equations, we have (obvious) expressions of MLDEs as: \[ L(f) \,=\,\mathfrak{d}_k^n(f)+\sum_{i=2}^nP_{2i}\mathfrak{d}_k^{n-i}(f) \] where $P_{2i}$ is a modular form of of weight $2i$ on $SL_2(\mathbb{Z})$ and $\mathfrak{d}_k(f)$ is the \textit{Serre derivative}. (We could replace $\Gamma$ by a Fuchsian subgroup of $SL_2(\mathbb{R})$ and allow the modular forms $P_{2i}$ to be meromorphic.) However, the iterated Serre derivative $\mathfrak{d}_k^n(f)$ (called a ``higher Serre derivation'' because as an operator it preserves modulality) is very complicated since it involves the Eisenstein series $E_2$. MLDEs, of course, can be given in the form % \[ % \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f)\quad\text{where $D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}$.} % \] \[ \mathsf{L}(f) \,=\, D^n(f)+\sum_{i=1}^nQ_iD^i(f) \] where \[ D=\frac{1}{2\pi\sqrt{-1}}\frac{d}{d\tau}. \] Then it is not easy to know if the equation above is an MLDE except the fact that $Q_i$ are quasimodular forms. Very recently, Y.~Sakai and D.~Zagier (my collaborators) found formulas of $\mathsf{L}(f)$ by using the Rankin--Cohen products between $f$ and $g_i$. This is a modular form of weight $2i$, which is a linear function of the differential of~$Q_{j}$. Moreover, there are \textit{inversion formulas} which express $Q_i$ as a linear function of the derivatives of $g_{j}$. The most important fact is that the order $n$ and $n-1$ parts are equal to the so-called higher Serre derivative in the sense of Kaneko and Koike, where the group is $\Gamma_1$. (This holds for any Fuchsian group.) \\ Finally, the most important nature of my talk is that I will use a \textbf{blackboard} instead of \textbf{slides}ss.

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