On the Inner Product of Positive Quadratic Forms

Barnes, Edwin; Cohn, M. J.

doi:10.1112/jlms/s2-12.1.32

Cited by 13 publications

(6 citation statements)

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“…Apply the automorphic condition to 71, 72, 73 and f, this proves (1), (2), (3). Let (~)m(/)=(~g,,,(/) be the space of holomorphic functions O(z, w) on Hg • ~g such that: (01) O(z,w+,cn 1 +rl2) =O(.c,w)e-2mni( ...... +2tnlw) where…”

Section: Dimension Formula Gmentioning

confidence: 70%

On the Kodaira dimension of the moduli space of abelian varieties

Tai

1982

Invent Math

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Section: Dimension Formula Gmentioning

confidence: 70%

On the Kodaira dimension of the moduli space of abelian varieties

Tai

1982

Invent Math

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“…To prove the proposition, we may obviously assume that m (A) = 1, so that A G K. By the result in [7] already mentioned, m (A) = min l yAy taken over y Ç Z w \ Therefore, K = {B G C: l yBy ^ 1 for all y £ Z n '}. Using Proposition 1, p. 128 of [6], we see that (**) K = closed convex hull of {y l y: y G 7J 1 ').…”

Section: Proof the Corollary Follows Easily Since Y L Y £ M (A) If Lmentioning

confidence: 90%

On Eutactic Forms

Ash

1977

Can. j. math.

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Let (aij) = A be a positive definite n × n symmetric matrix with real entries. To it corresponds a positive definite quadratic form ƒ on Rn: ƒ(x) = txAx = ∑ aijXiXj for x any column vector in Rn. The set of values ƒ(y) for y in Zn — {0} has a minimum m (A) > 0 and the number of “minimal vectors“ y1, … , yr in Zn for which ƒ(yi) = m (A) is finite. By definition, ƒ and A are called eutactic if and only if there are positive numbers s1 ,… , sr such that

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“…The cones are defined to be the cones over the faces of the convex hull of N ∩ (C \ 0). By a result of Barnes and Cohn [BC76], the vertices of Conv N ∩ (C \ 0) (that is, the rays of τ perf ) are of the form a * 2 , where a * = a i f * i is an integral primitive (i.e. indivisible) nonzero element of Λ * .…”

Section: Toroidal Compactifications Of a Gmentioning

confidence: 99%

Extending the Torelli map to toroidal compactifications of Siegel space

Alexeev

Brunyate

2011

Invent. math.

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It has been known since the 1970s that the Torelli map M g → A g , associating to a smooth curve its Jacobian, extends to a regular map from the Deligne-Mumford compactification M g to the 2nd Voronoi compactification A vor g . We prove that the extended Torelli map to the perfect cone (1st Voronoi) compactification A perf g is also regular, and moreover A vor g and A perf g share a common Zariski open neighborhood of the image of M g . We also show that the map to the Igusa monoidal transform (central cone compactification) is not regular for g ≥ 9; this disproves a 1973 conjecture of Namikawa.

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On the Inner Product of Positive Quadratic Forms

Cited by 13 publications

References 0 publications

On the Kodaira dimension of the moduli space of abelian varieties

On the Kodaira dimension of the moduli space of abelian varieties

On Eutactic Forms

Extending the Torelli map to toroidal compactifications of Siegel space

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