Weil-Barsotti formula for $\mathbf{T}$-modules
Dawid E. Kędzierski, Piotr Krasoń
TL;DR
This work extends the Weil-Barsotti formula to the function-field setting for strictly pure ${\mathbf{t}}$-modules with no nilpotence by computing the dual ${\mathbf{t}}-module ${\Phi}^{\vee}=\operatorname{Ext}_{0,\tau}^1({\Phi},C)$ and its double dual ${\Phi}^{\vee\vee}$ via a refined ${\mathbf{t}}$-reduction approach. The main results establish exact sequences linking ${\operatorname{Ext}}^1_{\tau}({\Phi},C)$ to ${\Phi}^{\vee}$ and ${\mathbb G}_a^d$, and similarly for ${\operatorname{Ext}}^1_{\tau}({\Phi}^{\vee},C)$ with ${\Phi}$, culminating in a biduality isomorphism ${({\Phi}^{\vee})}^{\vee}\cong {\Phi}$. The authors show that when $\deg_\tau{\Phi}=n\ge 2$ and $N_{\Phi}=0$, these constructions yield a coherent ${\mathbf{t}}$-module structure, with the $d=1$ case recovering the classical Weil-Barsotti result for Drinfeld modules; a counterexample with nonzero nilpotent part demonstrates the necessity of the hypotheses. The work connects to Cartier–Nishi-type dualities and Weil pairings in the function-field setting and illustrates how a careful, block-wise reduction process can realize dualities beyond Drinfeld modules.
Abstract
In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_τ^1(E,C)$ 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 \tm module $Φ$ with the zero nilpotent matrix $N_Φ.$ For such a \tm module $Φ$ we explicitly compute its dual \tm module $Φ^{\vee}$ as well as its double dual $Φ^{{\vee}{\vee}}.$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. Głoch, D.E. K{\k e}dzierski, P. Kraso{ń} [ Algorithms for determination of \tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{\k e}dzierski and P. Kraso{ń} [On $\Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_Φ$ is non-zero.