Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences

Scholl, A. J.

doi:10.1007/bf01388656

Cited by 50 publications

(30 citation statements)

References 12 publications

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“…Surprisingly this fact is not well known. It appeared for the first time implicitly in the work of Coleman [13] on p-adic modular forms, and later in [11,[21][22][23]. A direct proof in the case of level one was given in [6].…”

Section: Weakly Holomorphic Modular Formsmentioning

confidence: 99%

“…n+2 /D n+1 M ! −n can be interpreted as a space of modular forms of the second kind [6,11,23]. Indeed, it is canonically isomorphic to the algebraic de Rham cohomology of the moduli stack of elliptic curves with certain coefficients, and in particular, admits an action by Hecke operators.…”

Section: Real Frobeniusmentioning

confidence: 99%

“…The reader will easily be able to generalise the results of this paper to the case of a general congruence subgroup using the results of [23]. 1 In an "Appendix", we explain how the existence of a complex multiplication on the motive of a cusp form implies an algebraicity constraint on the single-valued involution.…”

Section: Weak Harmonic Lifts and Mock Modular Forms Of Integral Weightmentioning

confidence: 99%

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A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Brown

2018

Res Math Sci

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We define canonical real analytic versions of modular forms of integral weight for the full modular group, generalising real analytic Eisenstein series. They are harmonic Maass waveforms with poles at the cusp, whose Fourier coefficients involve periods and quasi-periods of cusp forms, which are conjecturally transcendental. In particular, we settle the question of finding explicit 'weak harmonic lifts' for every eigenform of integral weight k and level one. We show that mock modular forms of integral weight are algebro-geometric and have Fourier coefficients proportional to n 1−k (a n + ρa n ) for n = 0, where ρ is the normalised permanent of the period matrix of the corresponding motive, and a n , a n are the Fourier coefficients of a Hecke eigenform and a weakly holomorphic Hecke eigenform, respectively. More generally, this framework provides a conceptual explanation for the algebraicity of the coefficients of mock modular forms in the CM case.

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Section: Weakly Holomorphic Modular Formsmentioning

confidence: 99%

Section: Real Frobeniusmentioning

confidence: 99%

Section: Weak Harmonic Lifts and Mock Modular Forms Of Integral Weightmentioning

confidence: 99%

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A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Brown

2018

Res Math Sci

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“…The study of modular forms for noncongruence subgroups was initiated by Atkin and Swinnerton-Dyer [ASD], and continued in a sequence of papers [Sc1,Sc3,Sc4,Sc5,Sc6] by Scholl. Compared with what we know for forms for congruence subgroups, the overall knowledge on the arithmetic properties of forms for noncongruence subgroups, however, is far from satisfactory. Let Γ be a noncongruence subgroup of SL 2 (Z) with finite index.…”

Section: Modular Forms For Noncongruence Subgroupsmentioning

confidence: 99%

“…Atkin and Swinnerton-Dyer [ASD] pioneered the research in this area; they laid foundations and made a remarkable observation on the congruence property of Fourier coefficients of cusp forms for noncongruence subgroups, which we call Atkin-Swinnerton-Dyer congruences. Scholl [Sc1] attaches to the space of cusp forms of a given weight for a noncongruence subgroup a compatible family of l-adic Galois representations and proves that the Fourier coefficients of all cusp forms in the space satisfy certain congruence relations arising from the characteristic polynomials of the Frobenius elements. In case the space is one-dimensional, Scholl's result implies the Atkin-Swinnerton-Dyer congruences.…”

Section: Introductionmentioning

confidence: 99%

Modular Forms For Noncongruence Subgroups

Li¹,

Лонг²,

Yang³

2005

Pure and Applied Mathematics Quarterly

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Elliptic Crystals and Modular Motives

Ogus

2001

Advances in Mathematics

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The goal of this paper is to investigate the crystalline properties of families of elliptic curves or, more precisely, to study the families of crystals (elliptic crystals) attached to families of elliptic curves.Elliptic crystals over a point can be defined and classified very simply in terms of semilinear algebra, and this is how we begin (1.1). Our definition is cast in the logarithmic setting, so that it applies also to semistable (degenerate) elliptic curves. Roughly speaking, an elliptic crystal over a (logarithmic) perfect field k of characteristic p is a free W-module E of rank two equipped with a Frobenius-linear endomorphism 8, a nilpotent linear endomorphism N, a two-step Hodge filtration A on k E, and an isomorphism tr: 4 2 E Ä W. These data are required to satisfy various compatibilities: for example, A 1 (k E) is the kernel of id k 8. When N is not zero, such a crystal is quite rigid, and in particular its canonical coordinates are very precisely determined (1.3). Liftings of elliptic crystals from k to its Witt ring are determined by specifying a lifting B of the Hodge filtration A of k E, and the classification is made explicit in (1.5).Our first main result (2.2) concerns the relationship between deformation theory and elliptic crystals on a curve. Let XÂk be the reduction modulo p of a smooth log curve YÂW over W, let (E, B) be an elliptic crystal on YÂW, and let (E, A) be the restriction of (E, B) to XÂk. Following Mochizuki [8] and Gunning [6], we shall say that E is indigenous if its Kodaira-Spencer mapping is an isomorphism, a condition which can be checked either on YÂW or on XÂk. If f: X$ Ä X is a morphism of smooth log curves over k then a lifting g: Theorem (2.2) shows that if p is odd and E is indigenous, the resulting map from the set of liftings of f to the set of liftings of the mod p Hodge filtration of f *E X is a bijection. As a consequence we show (reprising some of the results of Mochizuki) in Theorem (2.4) that (when the degree of A 1 E X is positive and p is odd) an indigenous bundle on XÂk determines a unique lifting YÂW, characterized by the fact

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Modular forms and de Rham cohomology; Atkin-Swinnerton-Dyer congruences

Cited by 50 publications

References 12 publications

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

A class of non-holomorphic modular forms III: real analytic cusp forms for $$\mathrm {SL}_2(\mathbb {Z})$$ SL 2 ( Z )

Modular Forms For Noncongruence Subgroups

Elliptic Crystals and Modular Motives

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