On the Kodaira dimension of the moduli space of abelian varieties

Tai, Yung-sheng

doi:10.1007/bf01389411

Cited by 84 publications

(68 citation statements)

References 6 publications

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“…Exactly as in [6] (cf. also [9]) we obtain, for I large Л2 (p), but because the cusp form we have chosen is special these poles do not occur. That is because Ф vanishes to high order (at least order n) at the cusps.…”

Section: If P > 11 Is a Prime Then There Exists A Nontrivial Cusp Formentioning

confidence: 53%

Moduli of Abelian surfaces with a $(1,p^2)$ polarisation

Gritsenko¹,

Sankaran²

1996

Izv. RAN. Ser. Mat.

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Section: If P > 11 Is a Prime Then There Exists A Nontrivial Cusp Formentioning

confidence: 53%

Moduli of Abelian surfaces with a $(1,p^2)$ polarisation

Gritsenko¹,

Sankaran²

1996

Izv. RAN. Ser. Mat.

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“…In this case the subgroup of G that preserves acts transitively on the generators (s 23 k 2 k 3 interchanges x 2 1 and x 2 3 ; s 24 k 2 k 4 interchanges x 2 1 and x 2 4 ; more simply, consider the symmetries of a genuinely black forked graph), so we need only consider ¼ id. Then e is as in (23) and KðÞ ¼ fx 2 3 ; x 2 4 g, and e ¼ x 2 2 þ ðx 2 À x 4 Þ 2 þ x 2 4 : mðÞ ¼ x 2 4 so ðÞ ¼ ðÞ ¼ 2. Disconnected type hx 2 1 ; x 2 2 ; ðx 3 À x 4 Þ 2 i.…”

Section: Curves Of Depthmentioning

confidence: 99%

“…In this case x 2 1 and x 2 2 are interchanged by k 1 and x 2 1 and ðx 3 À x 4 Þ 2 are interchanged by w 0 , both of which preserve , so it is enough to consider ¼ id. Then e is as in (23) and KðÞ ¼ fx 2 2 ; ðx 3 À x 4 Þ 2 g, and e ¼ 2x 2 3 þ 2x 2 4 : mðÞ ¼ ðx 3 À x 4 Þ 2 so ðÞ ¼ ðÞ ¼ 4. Ã COROLLARY 4.24.…”

Section: Curves Of Depthmentioning

confidence: 99%

The nef cone of toroidal compactifications of ${\cal A}_4$

Hulek

Sankaran

2004

Proc. London Math. Soc.

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The moduli space A g of principally polarised abelian g-folds is a quasiprojective variety. It has a natural projective compactification, the Satake compactification, which has bad singularities at infinity. By the method of toroidal compactification we can construct other compactifications with milder singularities, at the cost of some loss of uniqueness. Two popular choices of toroidal compactification are the Igusa and the Voronoi compactifications: these agree for g ≤ 3 but for g = 4 they are different. In this paper, we shall be mainly interested in the Voronoi compactification and A Vor 4 . The proofs are inductive in the sense that they involve a reduction to the cases g = 3 and g = 2, where comparable results already exist; but some new techniques are also necessary for the proof. However, the Voronoi compactification for g > 4 is rather complicated and for this reason we are not at present able to extend our results even to g = 5. We also show (Theorem I.15) that the canonical bundle on A Igu 4 (n) is ample for n ≥ 3. The paper is structured as follows. Section I covers the facts we need about the different toroidal compactifications that are available. We describe the Voronoi compactification, in particular, in some detail, and state the main results. In Section II we explain what is known about the partial compactification of Mumford, which we shall need later. In Section III we describe the fine structure of the Voronoi boundary in the case g = 4, which is largely a matter of understanding the behaviour over the lowest stratum of the Satake compactification A Sat 4 . The methods here are toric and much is deduced from the combinatorics of a single cone in a certain 10-dimensional real vector space. The main technical result is that each non-exceptional 1

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“…The "slope" of a divisor refers to k/µ. The existence of effective divisors with slope less than the slope of the canonical bundle was used in [31] [22] to show that A n is of general type for n ≥ 7. This theorem shows that an effective divisor has slope which is bounded below by:…”

Section: Theoremmentioning

confidence: 99%

Linear dependence among Siegel modular forms

Poor¹,

Yuen²

2000

Mathematische Annalen

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Theorems are given which describe when high enough vanishing at the cusps implies that a Siegel modular cusp form is zero. Formerly impractical computations become practical and examples are given in degree four. Vanishing order is described by kernels, a type of polyhedral convex hull. Subject Classification (1991): 11F46, 11F27, 11H55 Mathematics IntroductionThis paper extends to Siegel modular forms certain practical computational techniques available for modular forms on the upper half plane. Two modular forms are equal when enough of their Fourier coefficients agree; more generally, a linear dependence relation holds among modular forms when it holds among enough of their Fourier coefficients. For example, in [29] Schiemann shows that the theta series for two distinct classes of 4 × 4 integral positive definite quadratic forms are equal by showing that their first 375 Fourier coefficients agree. The type of theorem one requires is that a cusp form is zero if it vanishes to a sufficiently large order; in the above example the cusp form in question is given by the difference of the theta series. In the case of Siegel modular forms, Siegel provided a version of the following result for the full modular group.Theorem (Siegel). Let f ∈ S k n have the Fourier expansion f (Ω) = s>0 a s e(tr (sΩ)). The following conditions are equivalent.(1) f = 0.(2) For all s such that tr(s) ≤ κ n k 4π , we have a s = 0. (3) For all s such that tr(s) ≤ nµ n n 2 √ 3 k 4π , we have a s = 0. (4) For all s such that det(s) 1/n ≤ µ n n 2 √ 3 k 4π , we have a s = 0.

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On the Kodaira dimension of the moduli space of abelian varieties

Cited by 84 publications

References 6 publications

Moduli of Abelian surfaces with a $(1,p^2)$ polarisation

Moduli of Abelian surfaces with a $(1,p^2)$ polarisation

The nef cone of toroidal compactifications of ${\cal A}_4$

Linear dependence among Siegel modular forms

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