Theta operators, Goss polynomials, and $v$-adic modular forms

Papanikolas, Matthew A.; Zeng, Guchao

Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 729-753.

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.

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.

Classification : 11F52, 11F33, 11G09
Mots clés : Drinfeld modular forms, Goss polynomials, $v$-adic modular forms, hyperderivatives, false Eisenstein series

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

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

[3] Brownawell, W. Dale 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

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

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

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

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

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

[9] Goss, David von Staudt for ${\mathbf{F}}_q\left[T\right]$, Duke Math. J., Volume 45 (1978), pp. 885-910 | DOI | Zbl

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

[11] Goss, David Modular forms for ${\mathbf{F}}_r\left[T\right]$, J. Reine Angew. Math., Volume 317 (1980), pp. 16-39 | Zbl

[12] Goss, David $\pi$-adic Eisenstein series for function fields, Compos. Math., Volume 41 (1980) no. 1, pp. 3-38 | Zbl

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

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

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

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

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

[18] Papanikolas, Matthew A. Log-algebraicity on tensor powers of the Carlitz module and special values of Goss $L$-functions (in preparation)

[19] Petrov, Aleksandar $A$-expansions of Drinfeld modular forms, J. Number Theory, Volume 133 (2013) no. 7, pp. 2247-2266 | DOI | Zbl

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

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

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

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

[24] Vincent, Christelle On the trace and norm maps from ${\Gamma }_0\left(𝔭\right)$ to ${\mathrm{GL}}_2\left(A\right)$, J. Number Theory, Volume 142 (2014), pp. 18-43 | DOI | Zbl

[25] Vincent, Christelle Weierstrass points on the Drinfeld modular curve $X_0\left(𝔭\right)$, Res. Math. Sci., Volume 2 (2015) (Paper No. 10, 40 pp.) | DOI | Zbl