On a ring of modular forms related to the theta gradients map in genus 2

Fiorentino, Alessio

doi:10.1016/j.jalgebra.2013.04.032

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…where n is a generic positive integer, and Γ g (1) = Γ g . The group Γ g (n) is a normal subgroup of Γ g , and the quotient group Γ g,n ≡ Γ g /Γ g (n), which is known as finite Siegel modular group, has finite order [222,223]:…”

Section: Symplectic Modular Invariancementioning

confidence: 99%

Neutrino mass and mixing with discrete symmetry

2013

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This is a review article about neutrino mass and mixing and flavour model building strategies based on discrete family symmetry. After a pedagogical introduction and overview of the whole of neutrino physics, we focus on the PMNS mixing matrix and the latest global fits following the Daya Bay and RENO experiments which measure the reactor angle. We then describe the simple bimaximal, tri-bimaximal and golden ratio patterns of lepton mixing and the deviations required for a non-zero reactor angle, with solar or atmospheric mixing sum rules resulting from charged lepton corrections or residual trimaximal mixing. The different types of see-saw mechanism are then reviewed as well as the sequential dominance mechanism. We then give a mini-review of finite group theory, which may be used as a discrete family symmetry broken by flavons either completely, or with different subgroups preserved in the neutrino and charged lepton sectors. These two approaches are then reviewed in detail in separate chapters including mechanisms for flavon vacuum alignment and different model building strategies that have been proposed to generate the reactor angle. We then briefly review grand unified theories (GUTs) and how they may be combined with discrete family symmetry to describe all quark and lepton masses and mixing. Finally we discuss three model examples which combine an SU (5) GUT with the discrete family symmetries A 4 , S 4 and ∆(96).

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Section: Symplectic Modular Invariancementioning

confidence: 99%

Neutrino mass and mixing with discrete symmetry

2013

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Section: Symplectic Modular Invariancementioning

confidence: 99%

Neutrino mass and mixing with modular symmetry

Ding,

King

2024

Rep. Prog. Phys.

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This is a review article about neutrino mass and mixing and flavour model building strategies based on modular symmetry. After an introduction to neutrino mass and lepton mixing, we then turn to the main subject of this review, namely a pedagogical introduction to modular symmetry as a candidate for family symmetry, from the bottom-up point of view. After an informal introduction to modular symmetry, we introduce the modular group, and discuss its fixed points and residual symmetry, assuming supersymmetry throughout. We then introduce finite modular groups of level N and modular forms with integer or rational modular weights, corresponding to simple geometric groups or their double or metaplectic covers, including the most general finite modular groups and vector-valued modular forms, with detailed results for N=2,3,4,5. The interplay between modular symmetry and generalized CP symmetry is discussed, deriving CP transformations on matter multiplets and modular forms, highlighting the CP fixed points and their implications. In general, compactification of extra dimensions generally leads to a number of moduli, and modular invariance with factorizable and non-factorizable multiple moduli based on symplectic modular invariance and automorphic forms is reviewed. Modular strategies for understanding fermion mass hierarchies are discussed, including the weighton mechanism, small deviations from fixed points, and texture zeroes. Then examples of modular models are discussed based on single modulus A4 models, a minimal S′4 model of leptons (and quarks), and a multiple moduli model based on three S4 groups capable of reproducing the Littlest Seesaw model. We then extend the discussion to include Grand Unified Theories (GUTs) based on modular (flipped) SU(5) and SO(10). Finally we discuss top-down approaches, including eclectic flavour symmetry and moduli stabilisation.

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“…Proof. Using equation (15) defining the map ψ, the computations with Jacobian determinants done by Fiorentino [Fio11], and lemma 1, we get…”

Section: Modular Forms Of Genus 2 and Level 2 And Binary Invariants mentioning

confidence: 99%

On the Coble quartic

Grushevsky¹,

Manni²

2015

ajm

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We obtain a short explicit expression for the universal Coble quartic whose partial derivatives give the defining equations for the universal family of Kummer threefolds. The Coble quartic was recently determined completely in Ren, Sam, Schrader, and Sturmfels, where (Theorem 7.1a) it was computed explicitly, as a polynomial with 372060 monomials of bidegree (28, 4) in theta constants of the second order and theta functions of the second order, respectively. Our expression is in terms of products of theta constants with characteristics corresponding to Göpel systems, and is a polynomial with 134 terms. Our approach is based on the beautiful geometry studied by Coble and further investigated by Dolgachev and Ortland, and highlights the geometry and combinatorics of syzygetic octets of characteristics, the GIT quotient for 7 points in P 2 , and the corresponding representations of Sp(6, F 2 ). One new ingredient is the relationship of Göpel systems and Jacobian determinants of theta functions. In genus 2, we similarly obtain a short explicit equation for the universal Kummer surface, and relate modular forms of level two to binary invariants of six points on P 1 . Introduction.In this paper we provide an explicit description for the (universal) Coble quartic. The universal Coble quartic is an equation in 16 = 8 + 8 variables, such that its partial derivatives with respect to the second variables give the defining equations for the universal family of Kummer varieties. The variety defined by the universal Coble quartic is the moduli space of semistable vector bundles of rank 2 with trivial determinant on genus three curves cf. [NR87], and our work thus provides one of the very few instances where such a moduli space is described completely explicitly. Knowing the Coble quartic explicitly could thus be of further use in investigating the Hitchin integrable system and in particular the Hitchin connection for this case.More precisely, the Coble quartic is a bihomogeneous polynomial of bidegree (28, 4) in 16 = 8 + 8 variables, which we think of as a hypersurface in P 7 × P 7 . Here the first P 7 is the target of the embedding of the moduli space of abelian threefolds (with level structure, A 3 (2, 4)) by theta constants of the second order. The second P 7 is the projectivization of the space of sections of the bundle |2Θ| on the corresponding abelian threefold. We denote theta functions of the second order by

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On a ring of modular forms related to the theta gradients map in genus 2

Cited by 3 publications

References 15 publications

Neutrino mass and mixing with discrete symmetry

Neutrino mass and mixing with discrete symmetry

Neutrino mass and mixing with modular symmetry

On the Coble quartic

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