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.

Congruences between modular forms: Raising the level and dropping Euler factors

We discuss the relationship among certain generalizations of results of Hida, Ribet, and Wiles on congruences between modular forms. Hida’s result accounts for congruences in terms of the value of an L-function, and Ribet’s result is related to the behavior of the period that appears there. Wiles’ theory leads to a class number formula relating the value of the L-function to the size of a Galois cohomology group. The behavior of the period is used to deduce that a formula at “nonminimal level” is obtained from one at “minimal level” by dropping Euler factors from the L-function.

Modular organization of intrinsic connections associated with spectral tuning in cat auditory cortex

Many response properties in primary auditory cortex (AI) are segregated spatially and organized topographically as those in primary visual cortex. Intensive study has not revealed an intrinsic, anatomical organizing principle related to an AI functional topography. We used retrograde anatomic tracing and topographic physiologic mapping of acoustic response properties to reveal long-range (≥1.5 mm) convergent intrinsic horizontal connections between AI subregions with similar bandwidth and characteristic frequency selectivity. This suggests a modular organization for processing spectral bandwidth in AI.

Modular construction for function of a ribonucleoprotein enzyme: the catalytic domain of Bacillus subtilis RNase P complexed with B.subtilis RNase P protein

The bacterial RNase P holoenzyme catalyzes the formation of the mature 5′-end of tRNAs and is composed of an RNA and a protein subunit. Among the two folding domains of the RNase P RNA, the catalytic domain (C-domain) contains the active site of this ribozyme. We investigated specific binding of the Bacillus subtilis C-domain with the B.subtilis RNase P protein and examined the catalytic activity of this C-domain–P protein complex. The C-domain forms a specific complex with the P protein with a binding constant of ∼0.1 µM. The C-domain–P protein complex and the holoenzyme are equally efficient in cleaving single-stranded RNA (∼0.9 min–1 at pH 7.8) and substrates with a hairpin–loop 3′ to the cleavage site (∼40 min–1). The holoenzyme reaction is much more efficient with a pre-tRNA substrate, binding at least 100-fold better and cleaving 10–500 times more efficiently. These results demonstrate that the RNase P holoenzyme is functionally constructed in three parts. The catalytic domain alone contains the active site, but has little specificity and affinity for most substrates. The specificity and affinity for the substrate is generated by either the specificity domain of RNase P RNA binding to a T stem–loop-like hairpin or RNase P protein binding to a single-stranded RNA. This modular construction may be exploited to obtain RNase P-based ribonucleoprotein complexes with altered substrate specificity.

A modular misexpression screen in Drosophila detecting tissue-specific phenotypes.

Genetic screens in Drosophila have lead to the discovery of many genes important for patterning and signal transduction in diverse organisms. Traditionally, the phenotypic effects of loss-of-function mutations are analyzed. As an alternative way to link genes and function, I have developed a versatile misexpression screen in Drosophila, the first such screen in higher eukaryotes. The screen identifies genes that, when over- or misexpressed in a pattern of interest, give a specific phenotype or modulate an existing mutant phenotype. It is based on Gal4 transactivation of a mobile enhancer and promoter that "targets" random endogenous genes for expression. The modular design of the screen allows directed expression in any temporal or spatial pattern. When activated in the developing eye, 4% of target inserts gave dominant phenotypes. One insertion was in the gene encoding Ras GTPase-activating protein; its overexpression phenotype was strongly enhanced by a mutation in Ras1. Thus, biologically relevant phenotypes and genetic interactions are identified using this method. The screen is a powerful new tool for developmental genetics; similar approaches can also be developed for other organisms.

Modular cis-regulatory organization of developmentally expressed genes: two genes transcribed territorially in the sea urchin embryo, and additional examples.

The cis-regulatory systems that control developmental expression of two sea urchin genes have been subjected to detailed functional analysis. Both systems are modular in organization: specific, separable fragments of the cis-regulatory DNA each containing multiple transcription factor target sites execute particular regulatory subfunctions when associated with reporter genes and introduced into the embryo. The studies summarized here were carried out on the CyIIIa gene, expressed in the embryonic aboral ectoderm and on the Endo16 gene, expressed in the embryonic vegetal plate, archenteron, and then midgut. The regulatory systems of both genes include modules that control particular aspects of temporal and spatial expression, and in both the territorial boundaries of expression depend on a combination of negative and positive functions. In both genes different regulatory modules control early and late embryonic expression. Modular cis-regulatory organization is widespread in developmentally regulated genes, and we present a tabular summary that includes many examples from mouse and Drosophila. We regard cis-regulatory modules as units of developmental transcription control, and also of evolution, in the assembly of transcription control systems.