Tomohide Terasoma

Mixed Tate motives and multiple zeta values

In this paper, we construct an object of the abelian category of mixed Tate motive associated to multiple zeta values. as a consequence, we prove the inequality of the dimension of the vector space generated by multiple zeta values, which is conjectured by Zagier. (Some errors was corrected from the first version. Namely, we changed the definition of the boundary

Theta constants associated to cubic threefolds

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Determinant of period integrals

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Period map of a certain K3 family with an 5-action

For a cubic surface X, by considering the intermediate Jacobian J(Y) of the triple covering Y of the 3-dimensional projective space branching along X, Allcock, Carlson and Toledo constructed a period map per from the family of marked cubic surfaces to the four dimensional complex ball embedded in the Siegel upper half space of degree 5. We give an expression of the inverse of per in terms of theta constants by constructing an isomorphism between J(Y) and a Prym variety of a cyclic 6-ple covering of the projective line branching at seven points.

Thomae type formula for K3 surfaces given by double covers of the projective plane branching along six lines

We prove a formula for the determinant of period integrals. Period integrals arise from comparison between Betti cohomologies and de Rham cohomologies. Our formula Theorem 1 in Section 4 expresses the determinant of period integrals as the product of the periods evaluated at the relative canonical cycles and special values of the Γ-function. The formula is a Hodge version of Theorem 1 of [S2] for `-adic cohomology. Together with this, it gives a motivic formula, Theorem 2 in Section 5. It particularly implies that the category of motives of rank 1 associated to an algebraic Hecke character is closed under taking the determinant of cohomology. Hence it gives a support to a conjecture of Deligne, Conjecture 8.1 iii [D4]; a motive of rank 1 is associated to an algebraic Hecke character. A typical example of our formula is that the period of a Fermat hypersurface is a product of special values of the Γ-function (cf. proof of Lemma 5.4). The main theorem for X = P is a reformulation of a theorem of the second named author [T], Theorem 1.2. The theorem is proved by reducing to this case by induction on dimension using a Lefschetz pencil. We describe the result. Let U be a smooth quasi-projective variety over a subfield k of C and F another subfield of C. The determinant of the periods, the main subject of the paper, is defined for a triple M = ((E ,∇), V, ρ) consisting of the following three data: (1) E is a locally free sheaf on U with an integrable connection ∇ regular singular along the boundary. It gives rise to a local system

ARITHMETIC-GEOMETRIC MEANS FOR HYPERELLIPTIC CURVES AND CALABI–YAU VARIETIES

Abstract In this paper, we study the period map of a certain one-parameter family of quartic K3 surfaces with an 𝔖5-action. We construct automorphic forms on the period domain as the pull-backs of theta constants of genus 2 by a modular embedding. Using these automorphic forms, we give an explicit presentation of the inverse period map.

Theta constants associated to coverings of P 1 branching at eight points

Abstract In this paper, we give a Thomae type formula for K3 surfaces X given by double covers of the projective plane branching along six lines. This formula gives relations between theta constants on the bounded symmetric domain of type I22 and period integrals of X. Moreover, we express the period integrals by using the hypergeometric function FS of four variables. As applications of our main theorem, we define ℝ4-valued sequences by mean iterations of four terms, and express their common limits by the hypergeometric function FS.

Algebraic correspondences between genus three curves and certain Calabi-Yau varieties

In this paper, we define a generalized arithmetic-geometric mean μg among 2g terms motivated by 2τ-formulas of theta constants. By using Thomaes formula, we give two expressions of μg when initial terms satisfy some conditions. One is given in terms of period integrals of a hyperelliptic curve C of genus g. The other is by a period integral of a certain Calabi–Yau g-fold given as a double cover of the g-dimensional projective space Pg.

Complete intersections with middle picard number 1 defined over Q

We construct automorphic forms on the 5-dimensional complex ball which give the inverse of the period map for cyclic 4-ple coverings of the complex projective line branching at eight points with branching index (1/4,...,1/4).

Selberg integrals and multiple zeta values

In this paper, we construct certain algebraic correspondences between genus three curves and certain type of Calabi-Yau threefolds which are double coverings of three dimensional projective spaces. Via this correspondence, the first cohomology groups of the curves can be embedded into the third cohomology groups of the Calabi-Yau three folds. Moreover we prove that the cokernel of this inclusion of variations of Hodge structures cannot be a factor of any variations of Hodge structures comming from polarized abelian schemes.