Search the site

Table of contents for this issue | Previous article | Next article
Matthew A. Papanikolas; Guchao Zeng
Theta operators, Goss polynomials, and $v$-adic modular forms
Journal de théorie des nombres de Bordeaux, 29 no. 3 (2017), p. 729-753, doi: 10.5802/jtnb.999
Article PDF
Class. Math.: 11F52, 11F33, 11G09
Keywords: Drinfeld modular forms, Goss polynomials, $v$-adic modular forms, hyperderivatives, false Eisenstein series

Résumé - Abstract

We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that $v$-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves $v$-integrality.


[1] Vincent Bosser & Federico Pellarin, Hyperdifferential properties of Drinfeld quasi-modular forms, Int. Math. Res. Not. 2008 (2008)
[2] Vincent Bosser & Federico Pellarin, On certain families of Drinfeld quasi-modular forms, J. Number Theory 129 (2009), p. 2952-2990 Article
[3] W. Dale Brownawell, Linear independence and divided derivatives of a Drinfeld module. I, Number Theory in Progress, Vol. 1 (Zakopane-Kościelisko, 1997), de Gruyter, 1999, p. 47–61
[4] Leonard Carlitz, An analogue of the von Staudt-Clausen theorem, Duke Math. J. 3 (1937), p. 503-517 Article
[5] Keith Conrad, The digit principle, J. Number Theory 84 (2000), p. 230-257 Article
[6] Jean Fresnel & Marius van der Put, Rigid Analytic Geometry and its Applications, Progress in Mathematics 218, Birkhäuser, 2004
[7] Ernst-Ulrich Gekeler, On the coefficients of Drinfeld modular forms, Invent. Math. 93 (1988), p. 667-700 Article
[8] Ernst-Ulrich Gekeler, On the zeroes of Goss polynomials, Trans. Am. Math. Soc. 365 (2013), p. 1669-1685 Article
[9] David Goss, von Staudt for $\mathbf{F}_q[T]$, Duke Math. J. 45 (1978), p. 885-910 Article
[10] David Goss, The algebraist’s upper half-plane, Bull. Am. Math. Soc. 2 (1980), p. 391-415 Article
[11] David Goss, Modular forms for $\mathbf{F}_r[T]$, J. Reine Angew. Math. 317 (1980), p. 16-39
[12] David Goss, $\pi $-adic Eisenstein series for function fields, Compos. Math. 41 (1980), p. 3-38
[13] David Goss, Basic Structures of Function Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, 35, 1996
[14] David Goss, A construction of $\mathfrak{v}$-adic modular forms, J. Number Theory 136 (2014), p. 330-338 Article
[15] Sangtae Jeong, Calculus in positive characteristic $p$, J. Number Theory 131 (2011) Article
[16] Bartolomé López, A non-standard Fourier expansion for the Drinfeld discriminant function, Arch. Math. 95 (2010), p. 143-150 Article
[17] Bartolomé López, Action of Hecke operators on two distinguished Drinfeld modular forms, Arch. Math. 97 (2011), p. 423-429 Article
[18] Matthew A. Papanikolas, “Log-algebraicity on tensor powers of the Carlitz module and special values of Goss $L$-functions”, in preparation
[19] Aleksandar Petrov, $A$-expansions of Drinfeld modular forms, J. Number Theory 133 (2013), p. 2247-2266 Article
[20] Aleksandar Petrov, On hyperderivatives of single-cuspidal Drinfeld modular forms with $\mathcal{A}$-expansions, J. Number Theory 149 (2015), p. 153-165 Article
[21] Jean-Pierre Serre, Formes modulaires et fonctions zêta $p$-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Lecture Notes in Math. 350, Springer, 20173, p. 191–268
[22] Yukiko Uchino & Takakazu Satoh, Function field modular forms and higher-derivations, Math. Ann. 311 (1998), p. 439-466 Article
[23] Christelle Vincent, Drinfeld modular forms modulo $\mathfrak{p}$, Proc. Am. Math. Soc. 138 (2010), p. 4217-4229 Article
[24] Christelle Vincent, On the trace and norm maps from $\Gamma _0(\mathfrak{p})$ to $\mathrm{GL}_2(A)$, J. Number Theory 142 (2014), p. 18-43 Article
[25] Christelle Vincent, Weierstrass points on the Drinfeld modular curve $X_0(\mathfrak{p})$, Res. Math. Sci. 2 (2015) Article