Uniruledness of orthogonal modular varieties

Gritsenko, Valery; Hulek, Klaus

doi:10.1090/s1056-3911-2014-00632-9

Cited by 27 publications

(27 citation statements)

References 25 publications

(35 reference statements)

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“…This theorem is a direct corollary of [7,Theorem 3.2] where it was proved that For ℓ = 1 our arguments also work. The orthogonal complement of D 7 in D 8 is the lattice D 1 = 4 of rank 1.…”

mentioning

confidence: 55%

Borcherds Products Everywhere

Gritsenko

Poor

Yuen

2015

Journal of Number Theory

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Abstract. We prove the Borcherds Products Everywhere Theorem, Theorem 6.6, that constructs holomorphic Borcherds Products from certain Jacobi forms that are theta blocks without theta denominator. The proof uses generalized valuations from formal series to partially ordered abelian semigroups of closed convex sets. We present nine infinite families of paramodular Borcherds Products that are simultaneously Gritsenko lifts. This is the first appearance of infinite families with this property in the literature.

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confidence: 55%

Borcherds Products Everywhere

Gritsenko

Poor

Yuen

2015

Journal of Number Theory

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“…The next theorem is a particular case of a more general result proved in a forthcoming paper by Gritsenko and a generalisation of [, Theorem 3.4]. Theorem Let

K

be a primitive sublattice of

N (R)

containing a direct summand of the same rank of a root lattice

R

of a Niemeier lattice

N (R)

or a primitive sublattice of the Leech lattice

N false(\emptyset false) = {normalΛ}_{24}

.…”

Section: The Strongly Reflective Modular Formsmentioning

confidence: 86%

“…(We note that in many cases

{\overset{normalO}{true}}^{+} false(2 U \oplus K false)

is not the maximal modular group of the reflective modular form.) The main theorem of gives a result on the geometric type of the corresponding modular varieties. Corollary For all 34 lattices

T

of signature

(n, 2)

from Theorem , Proposition and Theorem , the modular variety

{\overset{normalO}{true}}^{+} false(T false) ∖ Ω false(T false)

is at least uniruled.…”

Section: The Strongly Reflective Modular Formsmentioning

confidence: 99%

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Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

Gritsenko

Nikulin

2017

Proc. London Math. Soc.

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We describe a new large class of Lorentzian Kac–Moody algebras. For all ranks, we classify 2‐reflective hyperbolic lattices S with the group of 2‐reflections of finite volume and with a lattice Weyl vector. They define the corresponding hyperbolic Kac–Moody algebras of restricted arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac–Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identities are given by automorphic forms with 2‐reflective divisors. We give exact constructions of these automorphic forms as Borcherds products and, in some cases, as additive Jacobi liftings.

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“…We expect a long list of the moduli spaces K t of Kummer surfaces which are rational or unirational since the first cusp Γ * t -form of weight 3 is known only for t = 167 (see [19]). We note that the uniruledness of K t for a non-exceptional t = 21 was proved in [13].…”

Section: Introductionmentioning

confidence: 89%

Antisymmetric paramodular forms of weight 3

Gritsenko¹,

Wang²

2019

Sb. Math.

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The problem on the construction of antisymmetric paramodular forms of canonical weight 3 was open since 1998. Any cusp form of this type determines a canonical differential form on any smooth compactification of the moduli space of Kummer surfaces associated to (1, t)-polarised abelian surfaces. In this paper, we construct the first infinite family of antisymmetric paramodular forms of weight 3 as Borcherds products whose first Fourier-Jacobi coefficient is a theta block.

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Uniruledness of orthogonal modular varieties

Cited by 27 publications

References 25 publications

Borcherds Products Everywhere

Borcherds Products Everywhere

Lorentzian Kac-Moody algebras with Weyl groups of 2-reflections

Antisymmetric paramodular forms of weight 3

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