## On the zeroes of Goss polynomials

HTML articles powered by AMS MathViewer

- by Ernst-Ulrich Gekeler PDF
- Trans. Amer. Math. Soc.
**365**(2013), 1669-1685 Request permission

## Abstract:

Goss polynomials provide a substitute of trigonometric functions and their identities for the arithmetic of function fields. We study the Goss polynomials $G_k(X)$ for the lattice $A=\mathbb {F}_q[T]$ and obtain, in the case when $q$ is prime, an explicit description of the Newton polygon $NP(G_k(X))$ of the $k$-th Goss polynomial in terms of the $q$-adic expansion of $k-1$. In the case of an arbitrary $q$, we have similar results on $NP(G_k(X))$ for special classes of $k$, and we formulate a general conjecture about its shape. The proofs use rigid-analytic techniques and the arithmetic of power sums of elements of $A$.## References

- V. Bosser and F. Pellarin,
*Hyperdifferential properties of Drinfeld quasi-modular forms*, Int. Math. Res. Not. IMRN**11**(2008), Art. ID rnn032, 56. MR**2428858**, DOI 10.1093/imrn/rnn032 - Vincent Bosser and Federico Pellarin,
*On certain families of Drinfeld quasi-modular forms*, J. Number Theory**129**(2009), no. 12, 2952–2990. MR**2560846**, DOI 10.1016/j.jnt.2009.04.014 - Ernst-Ulrich Gekeler,
*Drinfel′d modular curves*, Lecture Notes in Mathematics, vol. 1231, Springer-Verlag, Berlin, 1986. MR**874338**, DOI 10.1007/BFb0072692 - Ernst-Ulrich Gekeler,
*On the coefficients of Drinfel′d modular forms*, Invent. Math.**93**(1988), no. 3, 667–700. MR**952287**, DOI 10.1007/BF01410204 - Ernst-Ulrich Gekeler,
*On power sums of polynomials over finite fields*, J. Number Theory**30**(1988), no. 1, 11–26. MR**960231**, DOI 10.1016/0022-314X(88)90023-6 - Ernst-Ulrich Gekeler,
*Zero distribution and decay at infinity of Drinfeld modular coefficient forms*, Int. J. Number Theory**7**(2011), no. 3, 671–693. MR**2805575**, DOI 10.1142/S1793042111004307 - Lothar Gerritzen and Marius van der Put,
*Schottky groups and Mumford curves*, Lecture Notes in Mathematics, vol. 817, Springer, Berlin, 1980. MR**590243** - David Goss,
*The algebraist’s upper half-plane*, Bull. Amer. Math. Soc. (N.S.)**2**(1980), no. 3, 391–415. MR**561525**, DOI 10.1090/S0273-0979-1980-14751-5 - David Goss,
*Basic structures of function field arithmetic*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Springer-Verlag, Berlin, 1996. MR**1423131**, DOI 10.1007/978-3-642-61480-4 - Jürgen Neukirch,
*Algebraic number theory*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR**1697859**, DOI 10.1007/978-3-662-03983-0 - Federico Pellarin,
*Aspects de l’indépendance algébrique en caractéristique non nulle (d’après Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu, et al.)*, Astérisque**317**(2008), Exp. No. 973, viii, 205–242 (French, with French summary). Séminaire Bourbaki. Vol. 2006/2007. MR**2487735** - http://oeis.org/wiki/Eulerian_polynomials.

## Additional Information

**Ernst-Ulrich Gekeler**- Affiliation: FR 6.1 Mathematik, Universität des Saarlandes, Postfach 15 11 50, D-66041 Saarbrücken, Germany
- Email: gekeler@math.uni-sb.de
- Received by editor(s): December 22, 2010
- Received by editor(s) in revised form: August 31, 2011
- Published electronically: October 15, 2012
- © Copyright 2012 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**365**(2013), 1669-1685 - MSC (2010): Primary 11F52; Secondary 11G09, 11J93, 11T55
- DOI: https://doi.org/10.1090/S0002-9947-2012-05699-6
- MathSciNet review: 3003278