scholarly journals Endomorphism ring of an induced module.

1965 ◽  
Vol 12 (2) ◽  
pp. 197-202 ◽  
Author(s):  
Patricia A. Tucker
Keyword(s):  
Endomorphism Ring ◽  
Induced Module

On Generalized Schur's Lemma and Its Applications

2012 ◽  
Vol 19 (02) ◽  
pp. 337-352 ◽  
Author(s):  
Lizhong Wang
Keyword(s):  
Representation Theory ◽  
Endomorphism Ring ◽  
Symmetric Groups ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
Endomorphism Rings ◽  
Natural Embedding ◽  
Induced Module ◽  
Schur’S Lemma ◽  
Schur's Lemma

In this paper, we generalize Schur's lemma on the basis of endomorphism rings for permutation modules. Let H be a subgroup of G and let M be a module of H. Set N = NG(H). Then there is a natural embedding of End N(MN) into End G(MG). By taking H to be a p-subgroup of G, we can reformulate Green's theory on modular representation. A defect theory is defined on the endomorphism ring of any induced module and it is used to prove Green's correspondence and related results. This defect theory can unify some well known results in modular representation theory. By using generalized Schur's lemma, we can also give a method to determine the multiplicity of simple modules in any permutation module of symmetric groups. This makes it possible to prove various versions of Foulkes' conjecture in a uniform way.

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

scholarly journals A trade-off between classical and quantum circuit size for an attack against CSIDH

2020 ◽  
Vol 15 (1) ◽  
pp. 4-17
Author(s):  
Jean-François Biasse ◽  
Xavier Bonnetain ◽  
Benjamin Pring ◽  
André Schrottenloher ◽  
William Youmans
Keyword(s):  
Endomorphism Ring ◽  
State Of The Art ◽  
Quantum Circuit ◽  
List Type ◽  
Hard Problem ◽  
Trade Off ◽  
Classical State ◽  
Security Parameter ◽  
The Cost ◽  
Quadratic Order

AbstractWe propose a heuristic algorithm to solve the underlying hard problem of the CSIDH cryptosystem (and other isogeny-based cryptosystems using elliptic curves with endomorphism ring isomorphic to an imaginary quadratic order 𝒪). Let Δ = Disc(𝒪) (in CSIDH, Δ = −4p for p the security parameter). Let 0 < α < 1/2, our algorithm requires:A classical circuit of size $2^{\tilde{O}\left(\log(|\Delta|)^{1-\alpha}\right)}.$A quantum circuit of size $2^{\tilde{O}\left(\log(|\Delta|)^{\alpha}\right)}.$Polynomial classical and quantum memory.Essentially, we propose to reduce the size of the quantum circuit below the state-of-the-art complexity $2^{\tilde{O}\left(\log(|\Delta|)^{1/2}\right)}$ at the cost of increasing the classical circuit-size required. The required classical circuit remains subexponential, which is a superpolynomial improvement over the classical state-of-the-art exponential solutions to these problems. Our method requires polynomial memory, both classical and quantum.

scholarly journals Cycles in the de Rham cohomology of abelian varieties over number fields

2018 ◽  
Vol 154 (4) ◽  
pp. 850-882
Author(s):  
Yunqing Tang
Keyword(s):  
Endomorphism Ring ◽  
Abelian Varieties ◽  
Number Fields ◽  
De Rham Cohomology ◽  
Tate Conjecture ◽  
De Rham ◽  
Prime Dimension

In his 1982 paper, Ogus defined a class of cycles in the de Rham cohomology of smooth proper varieties over number fields. This notion is a crystalline analogue of$\ell$-adic Tate cycles. In the case of abelian varieties, this class includes all the Hodge cycles by the work of Deligne, Ogus, and Blasius. Ogus predicted that such cycles coincide with Hodge cycles for abelian varieties. In this paper, we confirm Ogus’ prediction for some families of abelian varieties. These families include geometrically simple abelian varieties of prime dimension that have non-trivial endomorphism ring. The proof uses a crystalline analogue of Faltings’ isogeny theorem due to Bost and the known cases of the Mumford–Tate conjecture.

scholarly journals Fertility of Pedicellate Spikelets in Sorghum is Controlled by a Jasmonic Acid Regulatory Module

2019 ◽  
Author(s):  
Nicholas Gladman ◽  
Yinping Jiao ◽  
Young Koung Lee ◽  
Lifang Zhang ◽  
Ratan Chopra ◽  
...  
Keyword(s):  
Transcription Factor ◽  
Jasmonic Acid ◽  
Sorghum Bicolor ◽  
Spikelet Fertility ◽  
Cereal Crops ◽  
Grain Number ◽  
Regulatory Module ◽  
Induced Module ◽  
Acid Pathway ◽  
Floral Meristems

AbstractAs in other cereal crops, the panicles of sorghum (Sorghum bicolor (L.) Moench) comprise two types of floral spikelets (grass flowers). Only sessile spikelets (SSs) are capable of producing viable grains, whereas pedicellate spikelets (PSs) cease development after initiation and eventually abort. Consequently, grain number per panicle (GNP) is lower than the total number of flowers produced per panicle. The mechanism underlying this differential fertility is not well understood. To investigate this issue, we isolated a series of EMS-induced multiseeded (msd) mutants that result in full spikelet fertility, effectively doubling GNP. Previously, we showed that MSD1 is a TCP (Teosinte branched/Cycloidea/PCF) transcription factor that regulates jasmonic acid (JA) biosynthesis, and ultimately floral sex organ development. Here, we show that MSD2 encodes a lipoxygenase (LOX) that catalyzes the first committed step of JA biosynthesis. Further, we demonstrate that MSD1 binds to the promoters of MSD2 and other JA pathway genes. Together, these results show that a JA-induced module regulates sorghum panicle development and spikelet fertility. The findings advance our understanding of inflorescence development and could lead to new strategies for increasing GNP and grain yield in sorghum and other cereal crops.SignificanceThrough a single base pair mutation, grain number can be increased by ~200% in the globally important crop Sorghum bicolor. This mutation affects the expression of an enzyme, MSD2, that catalyzes the jasmonic acid pathway in developing floral meristems. The global gene expression profile in this enzymatic mutant is similar to that of a transcription factor mutant, msd1, indicating that disturbing any component of this regulatory module disrupts a positive feedback loop that occurs normally due to regular developmental perception of jasmonic acid. Additionally, the MSD1 transcription factor is able to regulate MSD2 in addition to other jasmonic acid pathway genes, suggesting that it is a primary transcriptional regulator of this hormone signaling pathway in floral meristems.

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