Adjoint modular Galois representations and their Selmer groups

In the last 15 years, many class number formulas and main conjectures have been proven. Here, we discuss such formulas on the Selmer groups of the three-dimensional adjoint representation ad(φ) of a two-dimensional modular Galois representation φ. We start with the p-adic Galois representation φ0 of a modular elliptic curve E and present a formula expressing in terms of L(1, ad(φ0)) the intersection number of the elliptic curve E and the complementary abelian variety inside the Jacobian of the modular curve. Then we explain how one can deduce a formula for the order of the Selmer group Sel(ad(φ0)) from the proof of Wiles of the Shimura–Taniyama conjecture. After that, we generalize the formula in an Iwasawa theoretic setting of one and two variables. Here the first variable, T, is the weight variable of the universal p-ordinary Hecke algebra, and the second variable is the cyclotomic variable S. In the one-variable case, we let φ denote the p-ordinary Galois representation with values in GL2(Zp[[T]]) lifting φ0, and the characteristic power series of the Selmer group Sel(ad(φ)) is given by a p-adic L-function interpolating L(1, ad(φk)) for weight k + 2 specialization φk of φ. In the two-variable case, we state a main conjecture on the characteristic power series in Zp[[T, S]] of Sel(ad(φ) ⊗ ν−1), where ν is the universal cyclotomic character with values in Zp[[S]]. Finally, we describe our recent results toward the proof of the conjecture and a possible strategy of proving the main conjecture using p-adic Siegel modular forms.

Genes Involved in Osmoregulation during Turgor-Driven Cell Expansion of Developing Cotton Fibers Are Differentially Regulated1

Cotton (Gossypium hirsutum L.) fibers are single-celled trichomes that synchronously undergo a phase of rapid cell expansion, then a phase including secondary cell wall deposition, and finally maturation. To determine if there is coordinated regulation of gene expression during fiber expansion, we analyzed the expression of components involved in turgor regulation and a cytoskeletal protein by measuring levels of mRNA and protein accumulation and enzyme activity. Fragments of the genes for the plasma membrane proton-translocating ATPase, vacuole-ATPase, proton-translocating pyrophosphatase (PPase), phosphoenolpyruvate carboxylase, major intrinsic protein, and α-tubulin were amplified by polymerase chain reaction and used as probes in ribonuclease protection assays of RNA from a fiber developmental series, revealing two discrete patterns of mRNA accumulation. Transcripts of all but the PPase accumulated to highest levels during the period of peak expansion (+12–15 d postanthesis [dpa]), then declined with the onset of secondary cell wall synthesis. The PPase was constitutively expressed through fiber development. Activity of the two proton-translocating-ATPases peaked at +15 dpa, whereas PPase activity peaked at +20 dpa, suggesting that all are involved in the process of cell expansion but with varying roles. Patterns of protein accumulation and enzyme activity for some of the proteins examined suggest posttranslational regulation through fiber development.