1-\(t\)-motifs. (English) Zbl 1441.11151
Böckle, Gebhard (ed.) et al., \(t\)-motives: Hodge structures, transcendence and other motivic aspects. EMS Series of Congress Reports. Zürich: European Mathematical Society (EMS). 417-439 (2020).
Summary: We show that the module of rational points on an abelian \(t\)-module \(E\) is canonically isomorphic with the module \(\text{Ext}^1(M_E, K[t])\) of extensions of the trivial \(t\)-motif \(K[t]\) by the \(t\)-motif \(M_E\) associated with \(E\). This generalizes prior results of G. W. Anderson and D. S. Thakur [Ann. Math. (2) 132, No. 1, 159–191 (1990; Zbl 0713.11082)], M. A. Papanikolas and N. Ramachandran [J. Number Theory 98, No. 2, 407–431 (2003; Zbl 1090.11040)], and S. S. Woo [Bull. Korean Math. Soc. 32, No. 2, 251–257 (1995; Zbl 0838.11042)].
In case \(E\) is uniformizable we show that this extension module is canonically isomorphic with the corresponding extension module of Pink-Hodge structures. This situation is formally very similar to Deligne’s theory of 1-motifs and we have tried to build up the theory in a way that makes this analogy as clear as possible.
For the entire collection see [Zbl 1441.14003].
In case \(E\) is uniformizable we show that this extension module is canonically isomorphic with the corresponding extension module of Pink-Hodge structures. This situation is formally very similar to Deligne’s theory of 1-motifs and we have tried to build up the theory in a way that makes this analogy as clear as possible.
For the entire collection see [Zbl 1441.14003].
MSC:
| 11G09 | Drinfel’d modules; higher-dimensional motives, etc. |
| 11-02 | Research exposition (monographs, survey articles) pertaining to number theory |
| 14A20 | Generalizations (algebraic spaces, stacks) |
| 11J93 | Transcendence theory of Drinfel’d and \(t\)-modules |
| 11M38 | Zeta and \(L\)-functions in characteristic \(p\) |
References:
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| [3] | Greg W. Anderson and Dinesh S. Thakur, Tensor powers of the Carlitz module and zeta values. Ann. Math. (2) 132(1) (1990), 159-191MR1059938. · Zbl 0713.11082 |
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| [9] | Matthew A. Papanikolas and Niranjan Ramachandran, A Weil-Barsotti formula for Drinfeld modules. J. Number Theory 98(2) (2003), 407-431.MR1955425. · Zbl 1090.11040 |
| [10] | Richard Pink, Hodge structures over function fields. pre-print, 1997. |
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