@MISC{Cadoret_kodairadimension, author = {Anna Cadoret}, title = {KODAIRA DIMENSION OF ABSTRACT MODULAR SURFACES}, year = {} }

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Abstract

Abstract. Let S be a variety over a field k of characteristic 0 with generic point η. Given a prime ℓ, a projective system ρ of rank-r Z/ℓ n-linear representations is a projective system of π1(S)-modules: (H ℓ n+1 → Hℓ n)n≥0 are induced by multipli-such that Hℓn = (Z/ℓn) r and the transition morphisms Hℓn+1 → Hℓn cation by ℓ. We will write: ρℓ n: π1(S) → GL(Hℓ n) = GLr(Z/ℓn) for the corresponding representations. To such data one can associate ‘abstract modular varieties’ Sρ,1(ℓ n) and Sρ(ℓ n) which, in this setting, are the modular analogues of the classical modular curves Y1(ℓ n) and Y (ℓ n). It is conjectured that, under some technical assumptions, Sρ,1(ℓ n) and Sρ(ℓ n) are of general type for n large enough. These technical assumptions are satisfied by Z/ℓ n-linear representations arising from the action of π1(S) on the étale cohomology groups with coefficients in Z/ℓ n of the geometric generic fiber of a smooth proper scheme over S. The case when S is a curve follows from previous works of A. Tamagawa and the author. In this paper, we consider the case when S is a surface. Our main result is that, under the technical assumptions mentioned above, Sρ(ℓ n) is of general type for n large enough. Also-under the same technical assumptions, we prove that Sρ,1(ℓ n) is of general type for n large enough provided S is neither rational, K3 nor Enriques. There is an arithmetic motivation for these geometric results. Indeed, if k is a number field and Sρ,1(ℓ n) is of general type, Lang conjecture predicts that Sρ,1(ℓ n)(k) is not Zariski-dense in Sρ,1(ℓ n). In particular, our result shows that Lang conjecture for surfaces implies that the k-rational ℓ-primary torsion in a family of (higher dimensional) abelian varieties parametrized by a surface S which is neither rational, K3 nor Enriques is uniformly bounded.