We investigate meromorphic quasi-modular forms and their L-functions. We study the space of meromorphic quasi-modular forms and Rankin–Cohen brackets of meromorphic modular forms. Then we define their L-functions by using multiple techniques of regularized integral. Explicit formulas for the L-functions are given.

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We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time singularities, and constructions of self-similar solutions.

We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the d -exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes [25]. Using the variation formalism, we prove that the relative periods of toric B-branes on a toric Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature [6], and discuss the relations between the works [25] [21] [6] in recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.

Let U be a connected non-singular quasi-projective variety and f : A → U a family of abelian varieties of dimension g. Suppose that the induced map U → Ag is generically finite and there is a compactification Y with complement S = Y \U a normal crossing divisor such that Ω1 Y (log S) is nef and ωY (S) is ample with respect to U. We characterize whether U is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map U →Ag or by the existence of CM points. More precisely, we show that f : A → U is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem V of R1f∗CA is either unitary or satisfies the Arakelov equality. Second, for each factor M in the universal cover of U whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M. If in addition f : A → U is rigid, it is a connected Shimura subvariety of Ag of Hodge type.