JGSP20(2010) 69–96
MODULAR FORMS ON BALL QUOTIENTS OF NON-POSITIVE KODAIRA DIMENSION
AZNIV KASPARIAN
Communicated by Vasil V. Tsanov
Abstract. The Baily-Borel compactification Bd/Γof an arithmetic ball quotient admits projective embeddings byΓ-modular forms of sufficiently large weight. We are interested in the target and the rank of the projective map Φ, determined by Γ-modular forms of weight one. This paper concentrates on the finiteH-Galois quotientsB/ΓH of a specificB/Γ(6,8)−1 , birational to an abelian surfaceA−1. Any compactification ofB/ΓH has non-positive Kodaira dimension. The rational maps ΦHofB\/ΓHare studied by means of theH-invariant abelian functions onA−1.
The modular forms of sufficiently large weight are known to provide projective embeddings of the arithmetic quotients of the two-ball
B={z= (z1, z2)∈C2;|z1|2+|z2|2 <1} 'SU(2,1)/S(U2×U1).
The present work studies the projective maps, given by the modular forms of weight one on certain Baily-Borel compactificationsB/Γ\H of Kodaira dimension κ(B/Γ\H) ≤ 0. More precisely, we start with a fixed smooth Picard modular surface A0−1 =
B/Γ(6,8)−1 0
with abelian minimal model A−1 = E−1 ×E−1, E−1 =C/Z+Zi. Any automorphism group ofA0−1, preserving the toroidal com- pactifying divisorT0 =
B/Γ(6,8)−1 0
\
B/Γ(6,8)−1
acts onA−1and lifts to a ball lat- ticeΓH, normalizingΓ(6,8)−1 . The ball quotient compactificationA0−1/H =B/ΓH is birational to A−1/H. We study theΓH-modular forms [ΓH,1]of weight one by realizing them asH-invariants of[Γ(6,8)−1 ,1]. That allows to transfer[ΓH,1]to theH-invariant abelian functions, in order to determinedimC[ΓH,1]and the tran- scendence dimension of the gradedC-algebra, generated by[ΓH,1]. The last one is exactly the rank of the projective mapΦ :B/Γ^H >P([ΓH,1]).
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