Diversity Arrays: a solid state technology for sequence information independent genotyping

Here we present the successful application of the microarray technology platform to the analysis of DNA polymorphisms. Using the rice genome as a model, we demonstrate the potential of a high-throughput genome analysis method called Diversity Array Technology, DArT‘. In the format presented here the technology is assaying for the presence (or amount) of a specific DNA fragment in a representation derived from the total genomic DNA of an organism or a population of organisms. Two different approaches are presented: the first involves contrasting two representations on a single array while the second involves contrasting a representation with a reference DNA fragment common to all elements of the array. The Diversity Panels created using this method allow genetic fingerprinting of any organism or group of organisms belonging to the gene pool from which the panel was developed. Diversity Arrays enable rapid and economical application of a highly parallel, solid-state genotyping technology to any genome or complex genomic mixtures.

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.