×

Algebraic independence of values of Goss \(L\)-functions at \(s=1\). (English) Zbl 1328.11082

A complete list of rings of algebraic functions over a finite field with a degree one rational place that have class number one is given in [J. R. C. Leitzel et al., J. Number Theory 7, 11–27 (1975; Zbl 0318.12009)]. Let \({\mathbb{F}}_q\) be a finite field with \(q\) elements, \(A_0={\mathbb{F}}_q[\theta]\) and \(A_1,A_2,A_3,A_4\) be the rings \({\mathbb{F}}_q[\theta,\eta]/(f_j)\) where, for \(j=1\), we have \(q=3\), \(f_1=\eta^2 - \theta^3 +\theta+1\in{\mathbb{F}}_3[\theta,\eta]\), for \(j=2\), we have \(q=4\), \(f_2 =\eta^2 +\eta+\theta^3 +\alpha\in{\mathbb{F}}_4[\theta,\eta], \alpha\in{\mathbb{F}}_4, \alpha^2 +\alpha+1=0\), for \(j=3\), we have \(q=2\), \(f_3 =\eta^2 +\eta+\theta^3 +\theta+1\in{\mathbb{F}}_2[\theta,\eta]\), for \(j=4\), we have \(q=2\), \(f_4 =\eta^2 +\eta+\theta^5 +\theta^3 +1\in{\mathbb{F}}_2[\theta,\eta]\). Let \(A\) be one of \(A_0,\dots,A_4\) and \(K\) the fraction field of \(A\). Set \(u=u_j\), where \(u_0=1/\theta\), \(u_j=\theta/\eta\) for \(j=1,2,3\) and \(u_4=\theta^2/\eta\). Let \(A_+\) be the set of monic elements in \(A\) and \({\mathcal P}\in A_+\) be an irreducible polynomial with residue field \(F_{\mathcal P}=A/{\mathcal P}\). Denote by \(\Xi_{\mathcal P}\) the group of all Dirichlet characters modulo \({\mathcal P}\) on \(A\). For \(\chi\in\Xi_{\mathcal P}\) and \(s\in{\mathbb{N}}\), consider the Goss \(L\)-function \[ L(s,\chi)=\sum_{a\in A_+} \frac{\chi(a)}{a^s} \in {\mathbb{F}}_{\mathcal P} ((u)). \] The main result of the paper under review is that the transcendence degree over \(K\) of the field generated by \(L(1,\chi)\), \(\chi\in\Xi_{\mathcal P}\), is \[ \frac{(q^d-1)(q-2)}{q-1}+1. \] Further, the authors give explicit formulae for the transcendental numbers \(L(1,\chi)\). The transcendence of this number was known in the special case of the trivial character \(\chi\) [D. S. Thakur, Int. Math. Res. Not. 1992, No. 9, 185–197 (1992; Zbl 0756.11015)], J. Yu [Invent. Math. 83, 507–517 (1986; Zbl 0586.12010)], and only in a few other cases G. Damamme [J. Théor. Nombres Bordx. 11, No. 2, 369–385 (1999; Zbl 0994.11027)].

MSC:

11J93 Transcendence theory of Drinfel’d and \(t\)-modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11M38 Zeta and \(L\)-functions in characteristic \(p\)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Anderson, G. W., Rank one elliptic \(A\)-modules and \(A\)-harmonic series, Duke Math. J., 73, 491-542 (1994) · Zbl 0807.11032
[2] Anderson, G. W., Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\), J. Number Theory, 60, 165-209 (1996) · Zbl 0868.11031
[3] Carlitz, L., On certain functions connected with polynomials in a Galois field, Duke Math. J., 1, 137-168 (1935) · JFM 61.0127.01
[4] Chang, C.-Y.; Papanikolas, M. A., Algebraic independence of periods and logarithms of Drinfeld modules. With an appendix by B. Conrad, J. Amer. Math. Soc., 25, 123-150 (2012) · Zbl 1271.11079
[5] Damamme, G., Étude de \(L(s, \chi) / \pi^s\) pour des fonctions \(L\) relatives à \(F_q((T^{- 1}))\) et associées à des caractères de degré 1, J. Théor. Nombres Bordeaux, 11, 369-385 (1999) · Zbl 0994.11027
[6] Feng, K., Andersonʼs root numbers and Thakurʼs Gauss sums, J. Number Theory, 65, 279-294 (1997)
[7] Goss, D., Basic Structures of Function Field Arithmetic (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0874.11004
[8] Hayes, D. R., Explicit class field theory in global function fields, (Studies in Algebra and Number Theory. Studies in Algebra and Number Theory, Adv. Math. Suppl. Stud., vol. 6 (1979), Academic Press: Academic Press New York), 173-217
[9] Hayes, D. R., A brief introduction to Drinfeld modules, (The Arithmetic of Function Fields. The Arithmetic of Function Fields, Columbus, OH, 1991 (1992), Walter de Gruyter: Walter de Gruyter Berlin), 1-32 · Zbl 0793.11015
[10] Leitzel, J. R.C.; Madan, M. L.; Queen, C. S., Algebraic function fields with small class number, J. Number Theory, 7, 11-27 (1975) · Zbl 0318.12009
[11] B.A. Lutes, Special values of the Goss \(L\) http://www.math.tamu.edu/ map/; B.A. Lutes, Special values of the Goss \(L\) http://www.math.tamu.edu/ map/
[12] Murty, M. R.; Murty, V. K., Transcendental values of class group \(L\)-functions, Math. Ann., 351, 835-855 (2011) · Zbl 1281.11071
[13] M.R. Murty, V.K. Murty, Transcendental values of class group \(L\) http://dx.doi.org/10.1090/S0002-9939-2012-11201-8; M.R. Murty, V.K. Murty, Transcendental values of class group \(L\) http://dx.doi.org/10.1090/S0002-9939-2012-11201-8 · Zbl 1282.11082
[14] Papanikolas, M. A., Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, Invent. Math., 171, 123-174 (2008) · Zbl 1235.11074
[15] Pellarin, F., \(τ\)-recurrent sequences and modular forms (2011)
[16] Taelman, L., Special \(L\)-values of Drinfeld modules, Ann. of Math. (2), 175, 369-391 (2012) · Zbl 1323.11039
[17] Thakur, D. S., Drinfeld modules and arithmetic in the function fields, Int. Math. Res. Not., 1992, 9, 185-197 (1992) · Zbl 0756.11015
[18] Thakur, D. S., Shtukas and Jacobi sums, Invent. Math., 111, 557-570 (1993) · Zbl 0770.11032
[19] Thakur, D. S., Function Field Arithmetic (2004), World Scientific Publishing: World Scientific Publishing River Edge, NJ · Zbl 1061.11001
[20] Yu, J., Transcendence and Drinfeld modules, Invent. Math., 83, 507-517 (1986) · Zbl 0586.12010
[21] Yu, J., Analytic homomorphisms into Drinfeld modules, Ann. of Math. (2), 145, 215-233 (1997) · Zbl 0881.11055
[22] Zhao, J., On root numbers connected with special values of \(L\)-functions over \(F_q(T)\), J. Number Theory, 62, 307-321 (1997) · Zbl 0874.11043
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.