On the coefficients of Drinfeld modular forms

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Formula (4.9) is an easy consequence of (2.8) and is analogous with Euler's formula {2ni)^'= —12^' n~^^. A proof of (4.10) may be found in [9], whereas

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" WalUs's formula" (4.11) is proved in [7]. Further, an old result of Wade, erably generalized by Yu [19], asserts the transcendence of n over K.

5 . Modular forms for GL (2, A)

( 5 . 1 ) Let us now consider the rank two case. A rank two Drinfeld module Ф is given by

where g and АФО are elements of С It corresponds to a rank two Л-lattice Y^^^^^ = Äoj^e)ÄW2mC.

DQrmoj=j { ф ) = g^^'/A.

Replacing Y by some similar lattice Я-7 (ЯеС*) will change (g, A) to (>l^"^g, À^~^^ A) but will leave j invariant. Thus, considering g. A, and j as functions of (û>i, CO2), we may restrict to pairs (со^, co2) = (z, 1). The discreteness condition on У translates to the condition "(coi, 0)2} X^o-linearly independent", i.e. z = coJoj2 lies in Q = Fi{C)^Fi{K^) = C^K^. On the "upper half-plane" Q, the group GL(2, K^) acts by fractional linear transformations

] { z ) = {az-\-b)/{cz-{-d). Let Y^ = Äz@Ä the lattice corresponding to zeQ.

\c d]

Two elements z,z' define similar lattices (i.e. isomorphic Drinfeld modules) if and only if they are equivalent by r = GL(2, A). Therefore,

( 5 . 2 ) r\Q = set of isomorphism classes of —^-^ С

rank two Drinfeld modules over С

ZI - - - - - - - - >j { z ) .

Next , we introduce the "imaginary part" \z\i oizeC. Put |z|f = inf|z-x| {xeK^\ and for с in the value group q^ of C,Qc={zsQ\\z\i^c}, Then |z|i = 0 is equivalent with zeK^, and an easy computation shows

( 5 . 3 ) |yzH|det7||cz + d|-2|zb

for y=(^ |eGL(2,Koo). We still have to say a few words about the analytic \c d)

structure on Q.

( 5 . 4 ) Proposition, (i) Q has the structure of a connected admissible open subspace ^fthe rigid analytic space Pi(C).

( ii ) Qc is an open admissible subspace ofQ.