Theta Constants and Teichmüller Modular Forms

Ichikawa, Takashi

doi:10.1006/jnth.1996.0156

Cited by 12 publications

(17 citation statements)

References 7 publications

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“…In this section we shall recall some notation and results from [14] and [19]. Let g 2 be an integer.…”

Section: Reduction Propertiesmentioning

confidence: 99%

Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Ritzenthaler¹

2010

LMS J. Comput. Math.

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Let k be a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A, a) over k, which is a Jacobian overk, being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

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“…In this section we shall recall some notation and results from [14] and [19]. Let g 2 be an integer.…”

Section: Reduction Propertiesmentioning

confidence: 99%

Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Ritzenthaler¹

2010

LMS J. Comput. Math.

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“…Ichikawa proved several important results on this modular form that we summarize in the following proposition, see [11,Prop.3.4] and [12]:…”

Section: Introductionmentioning

confidence: 93%

“…(C, λ)) which depends only on k-isomorphism class of the pair. With this definition, the following proposition holds, see for instance [11]: Proposition 1.1.1. The Torelli map t : M g −→ A g , associating to a curve C its Jacobian Jac C with the canonical polarization j, satisfies t * ω = λ, and induces for any field k a linear map…”

Section: Introductionmentioning

confidence: 99%

Jacobians among Abelian threefolds: A formula of Klein and a question of Serre

Lachaud

Ritzenthaler

Zykin

2010

Mathematical Research Letters

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Abstract. Let (A, a) be an indecomposable principally polarized abelian threefold defined over a field k ⊂ C. Using a certain geometric Siegel modular form χ 18 on the corresponding moduli space, we prove that (A, a) is a Jacobian over k if and only if χ 18 (A, a) is a square over k. This answers a question of J.-P. Serre. Then, via a natural isomorphism between invariants of ternary quartics and Teichmüller modular forms of genus 3, we obtain a simple proof of Klein formula, which asserts that χ 18 (Jac C, j) is equal to the square of the discriminant of C.

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“…The general definition of a Siegel modular form of degree g, level ℓ and weight k, defined over a Z-algebra M is a global section of µ ⊗k ⊗ Z M over X g (ℓ). Following T. Ichikawa [13], a Teichmüller modular form of degree g, level ℓ and weight k, defined over a Z-algebra M , is a global section of…”

Section: An Applicationmentioning

confidence: 99%

The Chiral Superstring Siegel Form in Degree Two Is a Lift

Poor

Yuen

2012

Journal of the Korean Mathematical Society

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Abstract. We prove that the Siegel modular form of D'Hoker and Phong that gives the chiral superstring measure in degree two is a lift. This gives a fast algorithm for computing its Fourier coefficients. We prove a general lifting from Jacobi cusp forms of half integral index t/2 over the theta group Γ 1 (1, 2) to Siegel modular cusp forms over certain subgroups Γ para (t; 1, 2) of paramodular groups. The theta group lift given here is a modification of the Gritsenko lift.

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Theta Constants and Teichmüller Modular Forms

Abstract: In this paper, we determine a primitive Teichmu ller modular form of degree g 3 over Z obtained from dividing the product of even theta constants by a certain integer, and we study this root as a Teichmu ller modular form over Q. 1996Academic Press, Inc.

Cited by 12 publications

References 7 publications

Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Jacobians among Abelian threefolds: A formula of Klein and a question of Serre

The Chiral Superstring Siegel Form in Degree Two Is a Lift

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