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p-adic Aspects of Jacobi Forms

✍ Scribed by Adriana Sofer

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 344 KB
Volume: 63
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1997.2095

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✦ Synopsis

We are interested in understanding and describing the p-adic properties of Jacobi forms. As opposed to the case of modular forms, not much work has been done in this area. The literature includes [3,4,7].

In the first section, we follow Serre's ideas from his theory of p-adic modular forms. We study Jacobi forms whose Fourier expansions have integral coefficients and look at congruences between them. Non-trivial examples are given by Jacobi Eisenstein series. It turns out that two Jacobi forms need to have the same index and satisfy a condition on the weights in order to be congruent.

If we define p-adic Jacobi forms in the natural way in this context, and restrict ourselves to the case of SL 2 (Z), we obtain a structure theorem for the space of p-adic Jacobi forms for SL 2 (Z) of a given weight / # Z$ p and index m # Z.

Another feature is that p-adic Jacobi forms for 1 0 ( p) are also forms for SL 2 (Z). This parallels the similar result for modular forms, and it will most probably play an important role in defining some p-adic operators that do not arise directly from complex operators.

In the second section, we associate to every Jacobi form with integral coefficients a measure on Z p with values in the p-adic ring of Katz's generalized modular forms. This is an injection that allows us to interpret Jacobi forms with p-adic coefficients as truly p-adic objects, and this suggests where to look for the adequate ``test objects'' for a modular p-adic theory. It also provides examples of p-adic analytic families of modular forms.

Finally, we point out that a lot of work remains to be done, starting by finding a modular definition of p-adic Jacobi forms and studying Hecke and other operators. We hope to eventually obtain some results on p-adic article no. NT972095 191 0022-314XÂ97 25.00

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