On modular Galois representations modulo prime powers

We study modular Galois representations mod $p^m$. We show that there are three progressively weaker notions of modularity for a Galois representation mod $p^m$: we have named these `strongly', `weakly', and `dc-weakly' modular. Here, `dc' stands for `divided congruence' in the sense of Katz and Hida. These notions of modularity are relative to a fixed level $M$. Using results of Hida we display a `stripping-of-powers of $p$ away from the level' type of result: A mod $p^m$ strongly modular representation of some level $Np^r$ is always dc-weakly modular of level $N$ (here, $N$ is a natural number not divisible by $p$). We also study eigenforms mod $p^m$ corresponding to the above three notions. Assuming residual irreducibility, we utilize a theorem of Carayol to show that one can attach a Galois representation mod $p^m$ to any `dc-weak' eigenform, and hence to any eigenform mod $p^m$ in any of the three senses. We show that the three notions of modularity coincide when $m=1$ (as well as in other, particular cases), but not in general

On the theta operator for modular forms modulo prime powers

We consider the classical theta operator $\theta$ on modular forms modulo $p^m$ and level $N$ prime to $p$ where $p$ is a prime greater than 3. Our main result is that $\theta$ mod $p^m$ will map forms of weight $k$ to forms of weight $k+2+2p^{m-1}(p-1)$ and that this weight is optimal in certain cases when $m$ is at least 2. Thus, the natural expectation that $\theta$ mod $p^m$ should map to weight $k+2+p^{m-1}(p-1)$ is shown to be false. The primary motivation for this study is that application of the $\theta$ operator on eigenforms mod $p^m$ corresponds to twisting the attached Galois representations with the cyclotomic character. Our construction of the $\theta$-operator mod $p^m$ gives an explicit weight bound on the twist of a modular mod $p^m$ Galois representation by the cyclotomic character

A multi-Frey approach to Fermat equations of signature (r,r,p)(r,r,p)

In this paper, we give a resolution of the generalized Fermat equations $$x^5 + y^5 = 3 z^n \text{ and } x^{13} + y^{13} = 3 z^n,$$ for all integers $n \ge 2$, and all integers $n \ge 2$ which are not a multiple of $7$, respectively, using the modular method with Frey elliptic curves over totally real fields. The results require a refined application of the multi-Frey technique, which we show to be effective in new ways to reduce the bounds on the exponents $n$. We also give a number of results for the equations $x^5 + y^5 = d z^n$, where $d = 1, 2$, under additional local conditions on the solutions. This includes a result which is reminiscent of the second case of Fermat's Last Theorem, and which uses a new application of level raising at $p$ modulo $p$.Comment: Includes more details regarding the connection of this paper with its sequel 'Some extensions of the modular method and Fermat-equations of signature (13,13,n)'. More precisely: extended Remark 7.4; added details on the computational parts of the proofs of Proposition 9 and Theorem 2; included new comments and polished the auxiliary Magma files for Proposition 9 and Theorem

A result on the equation xp+yp=zrx^p + y^p = z^r using Frey abelian varieties

We prove a diophantine result on generalized Fermat equations of the form $x^p + y^p = z^r$ which for the first time requires the use of Frey abelian varieties of dimension $\geq 2$ in Darmon's program. For that, we provide an irreducibility criterion for the mod $\mathfrak{p}$ representations attached to certain abelian varieties of $\text{GL}_2$-type over totally real fields

Nitrogen modulation of legume root architecture signaling pathways involves phytohormones and small regulatory molecules

Nitrogen, particularly nitrate is an important yield determinant for crops. However, current agricultural practice with excessive fertilizer usage has detrimental effects on the environment. Therefore, legumes have been suggested as a sustainable alternative for replenishing soil nitrogen. Legumes can uniquely form nitrogen-fixing nodules through symbiotic interaction with specialized soil bacteria. Legumes possess a highly plastic root system which modulates its architecture according to the nitrogen availability in the soil. Understanding how legumes regulate root development in response to nitrogen availability is an important step to improving root architecture. The nitrogen-mediated root development pathway starts with sensing soil nitrogen level followed by subsequent signal transduction pathways involving phytohormones, microRNAs and regulatory peptides that collectively modulate the growth and shape of the root system. This review focuses on the current understanding of nitrogen-mediated legume root architecture including local and systemic regulations by different N-sources and the modulations by phytohormones and small regulatory molecules.Nadiatul A. Mohd-Radzman was supported by ANU International PhD Scholarship. This work was supported by an Australian Research Council grant to Michael A. Djordjevic and Nijat Imin (DP140103714)

Diversification of the C-TERMINALLY ENCODED PEPTIDE (CEP) gene family in angiosperms, and evolution of plant-family specific CEP genes

BACKGROUND Small, secreted signaling peptides work in parallel with phytohormones to control important aspects of plant growth and development. Genes from the C-TERMINALLY ENCODED PEPTIDE (CEP) family produce such peptides which negatively regulate plant growth, especially under stress, and affect other important developmental processes. To illuminate how the CEP gene family has evolved within the plant kingdom, including its emergence, diversification and variation between lineages, a comprehensive survey was undertaken to identify and characterize CEP genes in 106 plant genomes. RESULTS Using a motif-based system developed for this study to identify canonical CEP peptide domains, a total of 916 CEP genes and 1,223 CEP domains were found in angiosperms and for the first time in gymnosperms. This defines a narrow band for the emergence of CEP genes in plants, from the divergence of lycophytes to the angiosperm/gymnosperm split. Both CEP genes and domains were found to have diversified in angiosperms, particularly in the Poaceae and Solanaceae plant families. Multispecies orthologous relationships were determined for 22% of identified CEP genes, and further analysis of those groups found selective constraints upon residues within the CEP peptide and within the previously little-characterized variable region. An examination of public Oryza sativa RNA-Seq datasets revealed an expression pattern that links OsCEP5 and OsCEP6 to panicle development and flowering, and CEP gene trees reveal these emerged from a duplication event associated with the Poaceae plant family. CONCLUSIONS The characterization of the plant-family specific CEP genes OsCEP5 and OsCEP6, the association of CEP genes with angiosperm-specific development processes like panicle development, and the diversification of CEP genes in angiosperms provides further support for the hypothesis that CEP genes have been integral to the evolution of novel traits within the angiosperm lineage. Beyond these findings, the comprehensive set of CEP genes and their properties reported here will be a resource for future research on CEP genes and peptides.We thank Jason Bragg for his input and advice on inferring gene trees. This work was supported by an Australian Research Council Discovery Project grant (DP120101893). HAO received financial support (UHS10488) to conduct this study from the Grains Research and Development Council