The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault, Park, Rülling, Krishna-Levine. Recently, Kerz-Saito introduced a notion of Chow group of 0-cycles with modulus in connection with geometric class field theory with wild ramification for varieties over finite fields. We study the non-homotopy invariant part of the Chow group of 0-cycles with modulus and show their torsion and divisibility properties.Modulus is being brought to sheaf theory by Kahn-Saito-Yamazaki in their attempt to construct a generalization of Voevodsky-Suslin-Friedlander's theory of homotopy invariant presheaves with transfers. We prove parallel results about torsion and divisibility properties for them.

We describe an algebra of meromorphic functions on the Siegel domain of genus two which contains Siegel modular forms for an arithmetic index six subgroup of the symplectic group and it is closed under three canonical derivations of the Siegel domain. The main ingredients of our study are the moduli of enhanced genus two curves, Gauss-Manin connection and the modular vector fields living on such moduli spaces.

In this paper, we study the general properties of commutative differential graded algebras in the category of representations over a reductive algebraic group with an injective central cocharacter. Besides describing the derived category of differential graded modules over such an algebra, we also provide a criterion for the existence of a t-structure on the derived category together with a characterization of the coordinate ring of the Tannakian fundamental group of its heart.

In this paper we describe the rigid tensor triangulated subcategory of Voevodsky's triangulated category of motives generated by the motive of an elliptic curve as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.1 For the CM case, we only consider the complex multiplication is defined over k.Furthermore, the piecewhere V (n,0) ⊗ V ⊗i ⊗V ⊗j are pairwise non-isomorphic irreducible representations over a K-algebra End Res K/Q Gm⊗K ((F K ) ⊗n ) and V i,j are pairwise non-isomorphic irreducible representation over Res K/Q G m ⊗ K = T K . For simplicity, we delete V i,j and one may think that both V andV are endowed with the G m -action. In fact, End Res K/Q Gm⊗K ((F K ) ⊗n ) is a special case defined in the Section 3.9 of [1], which is called B n,K . Ancona's main result -Theorem 4.1 in [1] implies that the decomposition like Lemma 2.1 is holding for the CM elliptic motives:3 Here λ t is the transpose (or conjugate) of λ, which is defined by interchanging rows and columns in the Young diagram associated to λ. 4 We view cn as an idempotent in End((F K ) ⊗n ), which lies in End Res K/Q Gm⊗K ((F K ) ⊗n ).

We consider the moduli space of abelian varieties with two marked points and a frame of the relative de Rham cohomolgy with boundary at these points compatible with its mixed Hodge structure. Such a moduli space gives a natural algebro-geometric framework for higher genus quasi Jacobi forms of index zero and their differential equations which are given as vector fields. In the case of elliptic curves we compute explicitly the Gauss-Manin connection and such vector fields.