On Atkin-Swinnerton-Dyer congruence relations

In this paper we exhibit a noncongruence subgroup \G whose space of weight 3 cusp forms S_3(\G) admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to S_3(\G) by A. Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.Comment: 25 page

On Atkin and Swinnerton-Dyer Congruence Relations (2)

In this paper we give an example of a noncongruence subgroup whose three-dimensional space of cusp forms of weight 3 has the following properties. For each of the four residue classes of odd primes modulo 8 there is a basis whose Fourier coefficients at infinity satisfy a three-term Atkin and Swinnerton-Dyer congruence relation, which is the $p$-adic analogue of the three-term recursion satisfied by the coefficients of classical Hecke eigen forms. We also show that there is an automorphic $L$-function over $\mathbb Q$ whose local factors agree with those of the $l$-adic Scholl representations attached to the space of noncongruence cusp forms.Comment: Last version, to appear on Math Annale

Fourier coefficients of noncongruence cuspforms

Given a finite index subgroup of SL2(ℤ) with modular curve defined over ℚ, under the assumption that the space of weight k (≥2) cuspforms is one-dimensional, we show that a form in this space with Fourier coefficients in ℚ has bounded denominators if and only if it is a congruence modular form. © 2012 London Mathematical Society

Draft Genome of the Leopard Gecko, \u3cem\u3eEublepharis Macularius\u3c/em\u3e

Background Geckos are among the most species-rich reptile groups and the sister clade to all other lizards and snakes. Geckos possess a suite of distinctive characteristics, including adhesive digits, nocturnal activity, hard, calcareous eggshells, and a lack of eyelids. However, one gecko clade, the Eublepharidae, appears to be the exception to most of these ‘rules’ and lacks adhesive toe pads, has eyelids, and lays eggs with soft, leathery eggshells. These differences make eublepharids an important component of any investigation into the underlying genomic innovations contributing to the distinctive phenotypes in ‘typical’ geckos. Findings We report high-depth genome sequencing, assembly, and annotation for a male leopard gecko, Eublepharis macularius (Eublepharidae). Illumina sequence data were generated from seven insert libraries (ranging from 170 to 20 kb), representing a raw sequencing depth of 136X from 303 Gb of data, reduced to 84X and 187 Gb after filtering. The assembled genome of 2.02 Gb was close to the 2.23 Gb estimated by k-mer analysis. Scaffold and contig N50 sizes of 664 and 20 kb, respectively, were compble to the previously published Gekko japonicus genome. Repetitive elements accounted for 42 % of the genome. Gene annotation yielded 24,755 protein-coding genes, of which 93 % were functionally annotated. CEGMA and BUSCO assessment showed that our assembly captured 91 % (225 of 248) of the core eukaryotic genes, and 76 % of vertebrate universal single-copy orthologs. Conclusions Assembly of the leopard gecko genome provides a valuable resource for future comptive genomic studies of geckos and other squamate reptiles

Computing Special LL-Values of Certain Modular Forms with Complex Multiplication

In this expository paper, we illustrate two explicit methods which lead to special $L$-values of certain modular forms admitting complex multiplication (CM), motivated in part by properties of $L$-functions obtained from Calabi-Yau manifolds defined over $\mathbb Q$