Theta operators, Goss polynomials, and v-adic modular forms
Matthew A. Papanikolas; Guchao Zeng
Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 3, p. 729-753

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.

Nous étudions les dérivées divisées des formes modulaires de Drinfeld et déterminons des formules pour ces dérivées en termes de polynômes de Goss pour le noyau de l’exponentielle de Carlitz. Comme conséquence, nous prouvons que les dérivées divisées des formes modulaires v-adiques au sens de Serre, définies par Goss et Vincent, sont encore des formes modulaires v-adiques. De plus, à multiplication par une factorielle de Carlitz près, la v-intégralité est stable sous les opérateurs de dérivation divisée.

Received : 2016-05-28
Accepted : 2016-12-02
Published online : 2017-12-13
DOI : https://doi.org/10.5802/jtnb.999
Classification:  11F52,  11F33,  11G09
Keywords: Drinfeld modular forms, Goss polynomials, v-adic modular forms, hyperderivatives, false Eisenstein series
@article{JTNB_2017__29_3_729_0,
     author = {Matthew A. Papanikolas and Guchao Zeng},
     title = {Theta operators, Goss polynomials, and $v$-adic modular forms},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     pages = {729-753},
     doi = {10.5802/jtnb.999},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/item/JTNB_2017__29_3_729_0}
}
Papanikolas, Matthew A.; Zeng, Guchao. Theta operators, Goss polynomials, and $v$-adic modular forms. Journal de Théorie des Nombres de Bordeaux, Volume 29 (2017) no. 3, pp. 729-753. doi : 10.5802/jtnb.999. https://jtnb.centre-mersenne.org/item/JTNB_2017__29_3_729_0/

[1] Vincent Bosser; Federico Pellarin Hyperdifferential properties of Drinfeld quasi-modular forms, Int. Math. Res. Not., Tome 2008 (2008) (Article ID rnn032, 56 pp.) | Zbl 1151.11021

[2] Vincent Bosser; Federico Pellarin On certain families of Drinfeld quasi-modular forms, J. Number Theory, Tome 129 (2009) no. 12, pp. 2952-2990 | Article | Zbl 1233.11052

[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), pp. 47-61 | Zbl 0931.11026

[4] Leonard Carlitz An analogue of the von Staudt-Clausen theorem, Duke Math. J., Tome 3 (1937), pp. 503-517 | Article | Zbl 63.0879.03

[5] Keith Conrad The digit principle, J. Number Theory, Tome 84 (2000) no. 2, pp. 230-257 | Article | Zbl 1017.11061

[6] Jean Fresnel; Marius van der Put Rigid Analytic Geometry and its Applications, Birkhäuser, Progress in Mathematics, Tome 218 (2004), xii+296 pages | Zbl 1096.14014

[7] Ernst-Ulrich Gekeler On the coefficients of Drinfeld modular forms, Invent. Math., Tome 93 (1988) no. 3, pp. 667-700 | Article | Zbl 0653.14012

[8] Ernst-Ulrich Gekeler On the zeroes of Goss polynomials, Trans. Am. Math. Soc., Tome 365 (2013) no. 3, pp. 1669-1685 | Article | Zbl 1307.11056

[9] David Goss von Staudt for F q [T], Duke Math. J., Tome 45 (1978), pp. 885-910 | Article | Zbl 0404.12013

[10] David Goss The algebraist’s upper half-plane, Bull. Am. Math. Soc., Tome 2 (1980), pp. 391-415 | Article | Zbl 0433.14017

[11] David Goss Modular forms for F r [T], J. Reine Angew. Math., Tome 317 (1980), pp. 16-39 | Zbl 0422.10021

[12] David Goss π-adic Eisenstein series for function fields, Compos. Math., Tome 41 (1980) no. 1, pp. 3-38 | Zbl 0422.10020

[13] David Goss Basic Structures of Function Field Arithmetic, 35, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 (1996), xiii+422 pages | Zbl 0874.11004

[14] David Goss A construction of 𝔳-adic modular forms, J. Number Theory, Tome 136 (2014), pp. 330-338 | Article | Zbl 1326.11020

[15] Sangtae Jeong Calculus in positive characteristic p, J. Number Theory, Tome 131 (2011) no. 6, 1089.1104 pages | Article | Zbl 1225.13027

[16] Bartolomé López A non-standard Fourier expansion for the Drinfeld discriminant function, Arch. Math., Tome 95 (2010) no. 2, pp. 143-150 | Article | Zbl 1227.11068

[17] Bartolomé López Action of Hecke operators on two distinguished Drinfeld modular forms, Arch. Math., Tome 97 (2011) no. 5, pp. 423-429 | Article | Zbl 1281.11046

[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, Tome 133 (2013) no. 7, pp. 2247-2266 | Article | Zbl 1286.11075

[20] Aleksandar Petrov On hyperderivatives of single-cuspidal Drinfeld modular forms with 𝒜-expansions, J. Number Theory, Tome 149 (2015), pp. 153-165 | Article | Zbl 06397313

[21] Jean-Pierre Serre Formes modulaires et fonctions zêta p-adiques, Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer (Lecture Notes in Math.) Tome 350 (20173), pp. 191-268 | Zbl 0277.12014

[22] Yukiko Uchino; Takakazu Satoh Function field modular forms and higher-derivations, Math. Ann., Tome 311 (1998) no. 3, pp. 439-466 | Article | Zbl 1155.11329

[23] Christelle Vincent Drinfeld modular forms modulo 𝔭, Proc. Am. Math. Soc., Tome 138 (2010) no. 12, pp. 4217-4229 | Article | Zbl 1228.11065

[24] Christelle Vincent On the trace and norm maps from Γ 0 (𝔭) to GL 2 (A), J. Number Theory, Tome 142 (2014), pp. 18-43 | Article | Zbl 1295.11049

[25] Christelle Vincent Weierstrass points on the Drinfeld modular curve X 0 (𝔭), Res. Math. Sci., Tome 2 (2015) (Paper No. 10, 40 pp.) | Article | Zbl 06587994