Explicit Construction of Rankin-Cohen-Type Differential Operators for Vector-Valued Siegel Modular Forms

Second-order integrable Lagrangians and WDVV equations

“…Here, we mainly follow [26,30], which specialised the general results of [17] to the genus two case. Let us introduce two matrix differential operators…”

“…Remark 5 It follows from [30], Proposition 2.3, that if f transforms as in (14), that is, as a weight −1 Siegel modular form, then the left-hand side of ( 16) transforms as a vector-valued Siegel modular form with values in the representation Sym 4 ⊗ det of GL(2, C).…”

“…Note that (30) can be obtained from ( 29) by differentiating it with respect to c and using ( 28), (29). Similarly, differentiating (30) with respect to c we obtain the Chazy equation [6] for h:…”

Second-order integrable Lagrangians and WDVV equations

“…This does not come as something unexpected, indeed, the integrability conditions possess Sp(4)-invariance ( 14) and, therefore, should be expressible via Sp(4)-invariant operations. Here we mainly follow [31,25], which specialised the general results of [19] to the genus two case. Let us introduce two matrix differential operators…”

“…Remark 5. It follows from [31], Proposition 2.3, that if f transforms as in (14), that is, as a weight −1 Siegel modular form, then the left-hand side of ( 16) transforms as a vector-valued Siegel modular form with values in the representation Sym 4 ⊗ det of GL(2, C).…”