[Paper Review] A characterization of certain Shimura curves in the moduli stack of abelian varieties

Eckart Viehweg, Kang Zuo|ArXiv.org|Jul 25, 2002

Algebraic Geometry and Number Theory18 references4 citations

TL;DR

This paper characterizes families of abelian varieties that achieve the Arakelov bound by showing they are isogenous to products of constant abelian varieties and modular elliptic curves, with the base curve being a Shimura curve. The maximality of the Higgs field in the Hodge bundle corresponds to special Hodge cycles and modular structures on the fibers, linking arithmetic geometry to Shimura varieties via variation of Hodge structures and monodromy constraints.

ABSTRACT

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the Arakelov inequalities say that 2deg(F(1,0)) is bounded from above by g=rank(F(1,0))(2q-2+s). We show that for s>0 families reaching this bound are isogenous to the g-fold product of a modular family of elliptic curves, and a constant abelian variety. For s=0, if the flat part of the VHS is defined over the rational numbers, the family is isogeneous to the g-fold product of a family h:Z-->Y, and a constant abelian variety. In this case, h:Z-->Y is obtained from the corestriction of a quaternion algebra A, defined over a totally real numberfield F, and ramified over all infinite places but one. In case the flat part of the VHS is not defined over the rational numbers, we determine the structure of the VHS.

Motivation & Objective

• To characterize families of abelian varieties that reach the Arakelov bound in terms of their Hodge-theoretic and geometric structure.
• To establish a link between maximality of the Higgs field and the existence of special Hodge cycles on the general fiber.
• To show that such families are isogenous to products of constant abelian varieties and modular elliptic curves over a finite étale cover.
• To connect the geometric and arithmetic properties of these families to the theory of Shimura varieties and Mumford-Tate groups.
• To provide an alternative to Shimura variety classification by using Higgs bundle decomposition and Simpson's correspondence.

Proposed method

• Use of the Deligne extension and Hodge filtration to construct the Higgs bundle (E, θ) from the variation of Hodge structures R¹f_*ℤ_X₀.
• Decomposition of the Higgs bundle into an ample part F and a flat part N via the decomposition theorem for Higgs bundles.
• Application of Simpson’s correspondence to relate the Higgs field maximality to the structure of local systems as direct sums and tensor products of rank-two weight-one variations and unitary systems.
• Analysis of monodromy and unipotency conditions to ensure the Higgs field is well-defined and compatible with the logarithmic structure on the base curve.
• Use of Arakelov inequalities and their equality case to characterize extremal families.
• Reduction to the case of modular elliptic curves via the maximality of the Higgs field, leading to the construction of Shimura curves as moduli spaces.

Experimental results

Research questions

• RQ1 What geometric and Hodge-theoretic conditions characterize families of abelian varieties that achieve the Arakelov bound?
• RQ2 How does the maximality of the Higgs field relate to the existence of special Hodge cycles on the general fiber?
• RQ3 Can families reaching the Arakelov bound be decomposed into simpler components, and if so, what are they?
• RQ4 What is the moduli-theoretic interpretation of such families in terms of Shimura varieties or totally geodesic subvarieties?
• RQ5 Are there families of Jacobians of curves that reach the Arakelov bound, and if so, under what conditions?

Key findings

• Families of abelian varieties reaching the Arakelov bound are isogenous over a finite étale cover to a product of a constant abelian variety and several copies of a modular elliptic curve family.
• The base curve Y is a Shimura curve, and the family arises as a pullback of the universal family over a moduli space with a suitable level structure.
• Maximality of the Higgs field (i.e., θ being an isomorphism) is equivalent to the Arakelov bound being achieved.
• For families with no unitary part in the local system, the Mumford-Tate group is defined over ℚ and the family is a pullback of the universal family over a Shimura variety with a given Hg subgroup.
• In the case of Jacobians of curves over ℙ¹, the Arakelov bound is achieved only if the number of singular fibers is exactly four, and such families are isogenous to products of constant abelian varieties and modular elliptic curves.
• Counterexamples exist showing that such families can arise as Jacobians, e.g., via Hurwitz covers of elliptic curves with level structures, with 3 additional singular fibers in the curve family beyond the cusps.