TOWARDS THE SIEGEL RING IN GENUS FOUR

2008 ◽  
Vol 04 (04) ◽  
pp. 563-586 ◽  
Author(s):  
MANABU OURA ◽  
CRIS POOR ◽  
DAVID S. YUEN
Keyword(s):  
Linear Combination ◽  
Irreducible Component ◽  
Modular Forms ◽  
Quotient Ring ◽  
Siegel Modular Forms ◽  
Weight Enumerators ◽  
New Techniques ◽  
First Order ◽  
The Difference ◽  
Infinite Set

Runge gave the ring of genus three Siegel modular forms as a quotient ring, R3/〈J(3)〉 where R3 is the genus three ring of code polynomials and J(3) is the difference of the weight enumerators for the e8 ⊕ e8 and [Formula: see text] codes. Freitag and Oura gave a degree 24 relation, [Formula: see text], of the corresponding ideal in genus four; where [Formula: see text] is also a linear combination of weight enumerators. We take another step towards the ring of Siegel modular forms in genus four. We explain new techniques for computing with Siegel modular forms and actually compute six new relations, classifying all relations through degree 32. We show that the local codimension of any irreducible component defined by these known relations is at least 3 and that the true ideal of relations in genus four is not a complete intersection. Also, we explain how to generate an infinite set of relations by symmetrizing first order theta identities and give one example in degree 32. We give the generating function of R5 and use it to reprove results of Nebe [25] and Salvati Manni [37].

Byte weight enumerators and modular forms of genus r

2015 ◽  
Vol 14 (06) ◽  
pp. 1550080
Author(s):  
Anuradha Sharma ◽  
Amit K. Sharma
Keyword(s):  
Modular Forms ◽  
Quadratic Forms ◽  
Zeta Functions ◽  
Siegel Modular Forms ◽  
Type I ◽  
Type Ii ◽  
Jacobi Forms ◽  
Weight Enumerators ◽  
Type Ii Codes ◽  
Real Quaternions

For a positive integer m, let R be either the ring ℤ2m of integers modulo 2m or the quaternionic ring Σ2m = ℤ2m + αℤ2m + βℤ2m + γℤ2m with α = 1 + î, β = 1 + ĵ and [Formula: see text], where [Formula: see text] are elements of the ring ℍ of real quaternions satisfying [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. In this paper, we obtain Jacobi forms (or Siegel modular forms) of genus r from byte weight enumerators (or symmetrized byte weight enumerators) in genus r of Type I and Type II codes over R. Furthermore, we derive a functional equation for partial Epstein zeta functions, which are summands of classical Epstein zeta functions associated with quadratic forms.

scholarly journals Higher pullbacks of modular forms on orthogonal groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Modular Forms ◽  
Differential Operators ◽  
Jacobi Form ◽  
Siegel Modular Forms ◽  
Orthogonal Groups

Abstract We apply differential operators to modular forms on orthogonal groups O ⁢ ( 2 , ℓ ) {\mathrm{O}(2,\ell)} to construct infinite families of modular forms on special cycles. These operators generalize the quasi-pullback. The subspaces of theta lifts are preserved; in particular, the higher pullbacks of the lift of a (lattice-index) Jacobi form ϕ are theta lifts of partial development coefficients of ϕ. For certain lattices of signature ( 2 , 2 ) {(2,2)} and ( 2 , 3 ) {(2,3)} , for which there are interpretations as Hilbert–Siegel modular forms, we observe that the higher pullbacks coincide with differential operators introduced by Cohen and Ibukiyama.

Integrating Sparse and Collaborative Representation Classifications for Image Classification

2017 ◽  
Vol 17 (02) ◽  
pp. 1750007 ◽  
Author(s):  
Chunwei Tian ◽  
Guanglu Sun ◽  
Qi Zhang ◽  
Weibing Wang ◽  
Teng Chen ◽  
...  
Keyword(s):  
Sparse Representation ◽  
Image Classification ◽  
Linear Combination ◽  
Image Recognition ◽  
Test Sample ◽  
Collaborative Representation ◽  
Training Samples ◽  
Sparse Representation Classification ◽  
The Difference ◽  
Representation Method

Collaborative representation classification (CRC) is an important sparse method, which is easy to carry out and uses a linear combination of training samples to represent a test sample. CRC method utilizes the offset between representation result of each class and the test sample to implement classification. However, the offset usually cannot well express the difference between every class and the test sample. In this paper, we propose a novel representation method for image recognition to address the above problem. This method not only fuses sparse representation and CRC method to improve the accuracy of image recognition, but also has novel fusion mechanism to classify images. The implementations of the proposed method have the following steps. First of all, it produces collaborative representation of the test sample. That is, a linear combination of all the training samples is first determined to represent the test sample. Then, it gets the sparse representation classification (SRC) of the test sample. Finally, the proposed method respectively uses CRC and SRC representations to obtain two kinds of scores of the test sample and fuses them to recognize the image. The experiments of face recognition show that the combination of CRC and SRC has satisfactory performance for image classification.

scholarly journals Lost chapters in CHL black holes: untwisted quarter-BPS dyons in the ℤ2 model

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Fabian Fischbach ◽  
Albrecht Klemm ◽  
Christoph Nega
Keyword(s):  
String Duality ◽  
Modular Forms ◽  
Black Hole Entropy ◽  
Siegel Modular Forms ◽  
Partition Functions ◽  
Twisted Sector ◽  
Duality Symmetry ◽  
Iwahori Subgroup ◽  
M Theory ◽  
Bonnet Term

Abstract Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $$ \mathcal{N} $$ N = 4 string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of “untwisted” quarter-BPS dyons in the heterotic ℤ2 CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup B(2) ⊂ Sp4(ℤ) which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold X = (K3 × T2)/ℤ2, as suggested by string duality with type IIA theory on X.

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