Compactification of Drinfeld moduli spaces as moduli spaces of $A$-reciprocal maps and consequences for Drinfeld modular forms

In this article we explain how the results in our previous article on ‘algebraic Hecke characters and compatible systems of mod$p$Galois representations over global fields’ allow one to attach a Hecke character to every cuspidal Drinfeld modular eigenform from its associated crystal that was constructed in earlier work of the author. On the technical side, we prove along the way a number of results on endomorphism rings of${\it\tau}$-sheaves and crystals. These are needed to exhibit the close relation between Hecke operators as endomorphisms of crystals on the one side and Frobenius automorphisms acting on étale sheaves associated to crystals on the other. We also present some partial results on the ramification of Hecke characters associated to Drinfeld modular eigenforms. An important phenomenon absent from the case of classical modular forms is that ramification can also result from places of modular curves of good but non-ordinary reduction. In an appendix, jointly with Centeleghe we prove some basic results on$p$-adic Galois representations attached to$\text{GL}_{2}$-type cuspidal automorphic forms over global fields of characteristic $p$.

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LINEAR RELATIONS FOR COEFFICIENTS OF DRINFELD MODULAR FORMS

International Journal of Number Theory ◽

10.1142/s1793042108001274 ◽

2008 ◽

Vol 04 (02) ◽

pp. 171-176

Author(s):

MATIJA KAZALICKI

Keyword(s):

Modular Forms ◽

Fourier Coefficients ◽

Linear Relations ◽

Drinfeld Modular Forms ◽

Divisibility Properties

Choie, Kohnen and Ono have recently classified the linear relations among the initial Fourier coefficients of weight k modular forms on SL2(ℤ), and they employed these results to obtain particular p-divisibility properties of some p-power Fourier coefficients that are common to all modular forms of certain weights. Using this, they reproduced some famous results of Hida on non-ordinary primes. Here we generalize these results to Drinfeld modular forms.

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