On Drinfeld modular forms of higher rank
We study Drinfeld modular forms for the modular group G GL(r,Fq[T on the Drinfeld symmetric space O, where r 2. Results include the existence of a (q 1) th root (up to constants) h of the discriminant function the description of the growth/decay of the standard forms g1, g2, gr 1, on the fundamental domain F of G, and the reduction of these forms on the central part Fo of F The results are exemplified in detail for r 3. Introduction Let F Fq be a finite field and A Fq[T be the polynomial ring in an indeterminate T with field of fractions K Fq(T Furthermore, K Fq((1/T is the completion of K at infinity, with completed algebraic closure C. The Drinfeld symmetric space Or P(C) where r 2, is acted upon by G GL(r,A) and the quotient G\Or parametrizes classes of A lattices L of rank r in C, that is, of Drinfeld modules of rank r. Such a Drinfeld module ph, corresponding to o Or, is given by an operator polynomial phT (X) TX g1X gr 1X r 1 grX r where the coefficients gi gi(o) depend on o, and the discriminant gr is nowhere zero. The dependence is such that the gi are modular forms for G, i.e. holomorphic, with a functional equation of the usual type under Manuscrit recu le 13 juin 2016, revise le 9 janvier 2017, accepte le 3 fevrier 2017. 2010 Mathematics Subject Classification. 11F52, 11G09, 14G22. Mots clefs. Drinfeld modular forms, Drinfeld discriminant function; Bruhat Tits building. 876 Ernst Ulrich Gekeler o 7 go (g G) and regular at infinity. For r 2, such Drinfeld modular forms (and their generalizations to congruence subgroups of G GL(2, A) were introduced by David Goss in his 1977 Harvard thesis and his papers [10, 11, 12] and further studied by the present author in the 1980's. The aim of this work is to generalize results known for r 2 (notably about the growth/decay of such forms, and the location of their zeroes) to larger ranks r. The plan of the paper is as follows. In the first section, we sketch the background on Drinfeld modules/modular forms and introduce notation. It doesn't contain any new material. In the second section, the relationship between Or and the Bruhat Tits building BT of PGL(r,K) is explained. This enables us to visualize the fundamental domain F Or for G via a standard Weyl chamber W in the realization BT (R) of BT We introduce the basic division functions mi (1 i r) in Section 3. The mi form an F basis of the T torsion of the generic Drinfeld module pho, where o runs through Or. They are modular forms of negative weight 1 for the congruence subgroup G(T of G, and are the key objects to get control over the gi and As a first consequence, we construct the form h, which satisfies hq 1 1) r T and is modular of weight (q r 1)(q 1) and type 1, see Theorem 3.8. The systematic study of the mi is given in Section 4. We give the increments of logq |mi(o) regarded as functions on the Weyl chamber W when k W (Z) is replaced by a neighboring vertex k' (Proposition 4.10) From this we deduce similar results for and the gi (Theorem 4.13 and its Corollaries 4.15 and 4.16) These results contain certain combinatorial numbers v k,i, which are investigated in the fifth section. We find an explicit and easy to evaluate expression in (5.3) which gives the final version Theorem 5.5 of Theorem 4.13 on the increments of logq (o) We also find the direction of largest descent of surprisingly, it strongly depends on the starting point k (Theorem 5.9) In Section 6 we study the behavior of g1, gr 1, gr on Fo (o1, or 1, 1) O |w1| |wr 1| 1} and the canonical reductions of the vanishing loci V (gi) Fo in Or(F) (Theorem 6.2) In particular, V (gi) Fo is non empty. In the final section, the case of r 3 is considered in more detail. Besides tables with values of some of the functions treated, we give a brief study of g1 at the wall F2 of F (where the zeroes of g1 are located) and of g2 at F1 (which encompasses the zeroes of g2) On Drinfeld modular forms of higher rank 877 Notation. F denotes throughout the finite field Fq with q elements, with algebraic closure F, and F(m) is the unique field extension of degree m of F in F. A F[T is the polynomial ring in an indeterminate T with field of fractions K F(T The completion at infinity of K is K F((p) with ring of integers O F[[p] where p T 1. We write C for the completed algebraic closure of K, OC for its ring of integers, and fix an identification of F with the residue class field of OC Then x 7 x is the canonical map from OC to F, with congruence relation x y x y. We normalize the absolute value onK by |T q and also write for its unique extension to C. log C Q is the map x 7 logq |x| and deg A }N0 is the degree map, with deg(0) with the usual conventions. For some fixed natural number r 2, G denotes the group scheme GL(r) with its center Z of scalar matrices, and G G(A) GL(r,A) (X) is the cardinality of the set X, G \X the space of G orbits of the group G that acts on X. I (resp. I) is the sum (or product) over the non zero elements of the index set I. (x1 xr) are projective coordinates in Pr 1; mostly we normalize xr 1; in this case we write (a1, ar 1, ar) (a1, ar 1, 1) for the corresponding point. 1. The basic set up (see e.g. [2] [5] [13, SS4] [16] A lattice in C is a discrete F subspace L of C, i.e. L intersects each ball in finitely many points. With such a L, we associate its lattice function eL C C, (1.1) eL(z) z