Weil-Barsotti formula for T-modules

In the work of M. A. Papanikolas and N. Ramachandran (2003) [26] the Weil-Barsotti formula for the function field case concerning ${\mathrm{Ext}}_{\tau }^1(E,C)$ where E is a Drinfeld module and C is the Carlitz module was proved. We generalize this formula to the case where E is a strictly pure t-module Φ with the zero nilpotent matrix $N_{\Phi }$. For such a t-module Φ we explicitly compute its dual t-module ${\Phi }^{\vee }$ as well as its double dual ${\Phi }^{\vee \vee }$. This computation is done in a subtle way by combination of the t-reduction algorithm developed by F. Głoch, D.E. Kędzierski, P. Krasoń, [arXiv:2408.08207] [13] and the methods of the work of D.E. Kędzierski and P. Krasoń (2024) [20]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_{\Phi }$ is non-zero.