Quasi-modular forms attached to elliptic curves: Hecke operators

Movasati, Hossein

doi:10.1016/j.jnt.2015.05.021

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…In the last row note that θ 4 4 = θ 4 3 − θ 4 2 . In the third row we have a polynomial relation between the three modular forms there, see for instance the last section of [Mov15c].…”

Section: Modular Forms and Gauss Hypergeometric Equationmentioning

confidence: 99%

Calabi-Yau modular forms in limit: Elliptic fibrations

Haghighat¹,

Movasati²,

Yau³

2017

Communications in Number Theory and Physics

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We study the limit of Calabi-Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi-Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi-Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi-Yau fourfolds we derive new identities not computed before.

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“…In the last row note that θ 4 4 = θ 4 3 − θ 4 2 . In the third row we have a polynomial relation between the three modular forms there, see for instance the last section of [Mov15c].…”

Section: Modular Forms and Gauss Hypergeometric Equationmentioning

confidence: 99%

Calabi-Yau modular forms in limit: Elliptic fibrations

Haghighat¹,

Movasati²,

Yau³

2017

Communications in Number Theory and Physics

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“…Since the rational homotopy groups of Σ −1 ko are concentrated in odd degrees and since the only odd-degree rational homotopy group of F is the copy of Q in rational π 3 (F ) coming from the failure of π 4 (tmf ) → π 4 (ko) to be rationally surjective, the dotted arrow in (4) is well-defined up to the question of how T n ought to act on π 3 (F ) ⊗ Z Q; a reasonable way to make this choice might be to use the action of T n on the quasimodular form E 2 , as in [Mov15]. For the sake of this note, one can make whatever choice one wants, since we make no use of the Hecke action on π 3 (F ); since taking the 4-connective cover is a functor, each map F ∧HQ → F ∧HQ yields a map tcf ∧HQ → tcf ∧HQ with the same effect on homotopy in degrees ≥ 4, yielding our Hecke action on tcf ∧HQ.…”

Section: Topological Cusp Formsmentioning

confidence: 99%

The topological Petersson product

Candelori¹,

Salch²

2022

Preprint

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The nondegeneracy of the Petersson inner product on cusp forms, and the fact that Hecke operators are self-adjoint with respect to the Petersson product, together imply that the cusp forms have a basis consisting of Hecke eigenforms. In the literature on topological modular forms, no topological analogue of the Petersson product is to be found, and it is not known which topological spaces have the property that their topological cusp forms admit a basis consisting of eigenforms for the action of Baker's topological Hecke operators. In this note we define and study a natural topological Petersson product on complexified topological cusp forms, whose value on a one-point space recovers the classical Petersson product. We find that the topological Petersson product is usually degenerate: in particular, if X is a space with nontrivial rational homology in any positive degree, then the Petersson product on complexified tcf * (X) is degenerate. Despite tcf -cohomology being a stable invariant, the topological Petersson product is an unstable invariant, vanishing on all suspensions. Nevertheless, we demonstrate nontriviality of the topological Petersson product by giving an explicit calculation of the topological Petersson product on complexified tcf -cohomology of the complex projective plane. We show that, for a compact Kahler manifold X, the Petersson product is nondegenerate on complexified tcf -cohomology of X in a range of Atiyah-Hirzebruch filtrations (essentially one-third of the possibly-nonzero Atiyah-Hirzebruch filtrations in the tcf -cohomology of X). Contents 1. Introduction 1 2. Topological cusp forms. 3 3. Hecke operators on rational topological cusp forms. 4 4. The Petersson product on complexified topological cusp forms. 6 References 12

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“…Mathematical experiments are cheap provided that one knows which kind of experiment to do. In this note we report such an experiment which arose from the first author's work on quasi-modular forms, see [Mov12,Mov15], the following simple equality (1.1)…”

Section: Introductionmentioning

confidence: 99%

Product formulas for weight two newforms

Movasati,

Nikdelan

2018

Preprint

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Quasi-modular forms attached to elliptic curves: Hecke operators

Cited by 3 publications

References 12 publications

Calabi-Yau modular forms in limit: Elliptic fibrations

Calabi-Yau modular forms in limit: Elliptic fibrations

The topological Petersson product

Product formulas for weight two newforms

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