## Prolongations of $t$-Motives and Algebraic Independence of Periods

##### Doc. Math. 23, 815-838 (2018)
DOI: 10.25537/dm.2018v23.815-838

### Summary

In this article we show that the coordinates of a period lattice generator of the $n$-th tensor power of the Carlitz module are algebraically independent, if $n$ is prime to the characteristic. The main part of the paper, however, is devoted to a general construction for $t$-motives which we call prolongation, and which gives the necessary background for our proof of the algebraic independence. Another incredient is a theorem which shows hypertranscendence for the Anderson-Thakur function $\omega(t)$, i.e. that $\omega(t)$ and all its hyperderivatives with respect to $t$ are algebraically independent.

### Mathematics Subject Classification

11J93, 11G09, 13N99

### Keywords/Phrases

Drinfeld modules, $t$-modules, transcendence, higher derivations, hyperdifferentials

### References

• 1. Greg W. Anderson. $t$-motives. Duke Math. J., 53(2):457--502, 1986. DOI 10.1215/S0012-7094-86-05328-7; zbl 0679.14001; MR0850546.
• 2. Greg W. Anderson, W. Dale Brownawell, and Matthew A. Papanikolas. Determination of the algebraic relations among special $\Gamma$-values in positive characteristic. Ann. of Math. (2), 160(1):237--313, 2004. DOI 10.4007/annals.2004.160.237; zbl 1064.11055; MR2119721; arxiv math/0207168.
• 3. Greg W. Anderson and Dinesh S. Thakur. Tensor powers of the Carlitz module and zeta values. Ann. of Math. (2), 132(1):159--191, 1990. DOI 10.2307/1971503; zbl 0713.11082; MR1059938.
• 4. Bruno Anglès and Federico Pellarin. Universal Gauss-Thakur sums and $L$-series. Invent. Math., 200(2):653--669, 2015. DOI 10.1007/s00222-014-0546-8; zbl 1321.11053; MR3338012; arxiv 1301.3608.
• 5. W. Dale Brownawell. Linear independence and divided derivatives of a Drinfeld module. I. In Number theory in progress, Vol. 1 (Zakopane-Kościelisko, 1997), pages 47--61. de Gruyter, Berlin, 1999. zbl 0931.11026; MR1689498.
• 6. W. Dale Brownawell. Minimal extensions of algebraic groups and linear independence. J. Number Theory, 90(2):239--254, 2001. DOI 10.1006/jnth.2001.2638; zbl 0997.11056; MR1858075.
• 7. W. Dale Brownawell and Laurent Denis. Linear independence and divided derivatives of a Drinfeld module. II. Proc. Amer. Math. Soc., 128(6):1581--1593, 2000. DOI h10.1090/S0002-9939-00-05633-1; zbl 1099.11509; MR1709742.
• 8. W. Dale Brownawell and Matthew A. Papanikolas. A rapid introduction to Drinfeld modules, t-modules, and t-motives. To appear in t-Motives: Hodge Structures, Transcendence, and Other Motivic Aspects. arxiv 1806.03919.
• 9. Chieh-Yu Chang and Matthew A. Papanikolas. Algebraic independence of periods and logarithms of Drinfeld modules. J. Amer. Math. Soc., 25(1):123--150, 2012. With an appendix by Brian Conrad. DOI 10.1090/S0894-0347-2011-00714-5; zbl 1271.11079; MR2833480; arxiv 1005.5120.
• 10. David Goss. Basic structures of function field arithmetic, volume 35 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1996. zbl 0874.11004; MR1423131.
• 11. Urs Hartl and Ann-Kristin Juschka. Pink's theory of Hodge structures and the Hodge conjecture over function fields. To appear in t-Motives: Hodge Structures, Transcendence, and Other Motivic Aspects. arxiv 1607.01412.
• 12. Moshe Kamensky. Tannakian formalism over fields with operators. Int. Math. Res. Not. IMRN, (24):5571--5622, 2013. DOI 10.1093/imrn/rns190; zbl 1335.12005; MR3144174; arxiv 1111.7285.
• 13. Hideyuki Matsumura. Commutative ring theory, volume 8 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid. zbl 0666.13002; MR1011461.
• 14. Andreas Maurischat and Rudolph Perkins. An integral digit derivative basis for Carlitz prime power torsion extension. Preprint available from arXiv at http://arxiv.org/abs/1611.09681, November 2016.
• 15. Yoshinori Mishiba. Algebraic independence of the Carlitz period and the positive characteristic multizeta values at $n$ and $(n,n)$. Proc. Amer. Math. Soc., 143(9):3753--3763, 2015. DOI 10.1090/S0002-9939-2015-12532-4; zbl 1322.11082; MR3359567; arxiv 1307.3725.
• 16. Matthew A. Papanikolas. Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms. Invent. Math., 171(1):123--174, 2008. DOI 10.1007/s00222-007-0073-y; zbl 1235.11074; MR2358057; arxiv math/0506078.
• 17. Federico Pellarin. On a variant of Schanuel conjecture for the Carlitz exponential. Theor. Nombres Bordx., 29(3):845-873, 2017. DOI 10.5802/jtnb.1004; zbl 06857228; arxiv 1610.04048.
• 18. L. I. Wade. Certain quantities transcendental over $GF(p^n,x)$. Duke Math. J., 8:701--720, 1941. MR0006157.

### Affiliation

Maurischat, Andreas
Lehrstuhl A für Mathematik, RWTH Aachen, University, Aachen, Germany