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

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Accepté le :
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DOI : https://doi.org/10.5802/jtnb.999
Classification : 11F52,  11F33,  11G09
Mots clés : 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},
     pages = {729--753},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {29},
     number = {3},
     year = {2017},
     doi = {10.5802/jtnb.999},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.999/}
}
Matthew A. Papanikolas; Guchao Zeng. Theta operators, Goss polynomials, and $v$-adic modular forms. Journal de Théorie des Nombres de Bordeaux, Tome 29 (2017) no. 3, pp. 729-753. doi : 10.5802/jtnb.999. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.999/

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