Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields

There is a natural probability measure on the set of isomorphism classes of principally polarized Abelian varieties of dimension $g$ over $\mathbb{F}_q$, weighted by the number of automorphisms. The distributions of the number of $\mathbb{F}_q$-rational points are related to the cohomology of fiber powers of the universal family of principally polarized Abelian varieties. To that end we compute the cohomology $H^i(\mathcal{X}^{\times n}_g,\mathbb{Q}_\ell)$ for $g=1$ using results of Eichler-Shimura and for $g=2$ using results of Lee-Weintraub and Petersen, and we compute the compactly supported Euler characteristics $e_\mathrm{c}(\mathcal{X}^{\times n}_g,\mathbb{Q}_\ell)$ for $g=3$ using results of Hain and conjectures of Bergstr\"om-Faber-van der Geer. In each of these cases we identify the range in which the point counts $\#\mathcal{X}^{\times n}_g(\mathbb{F}_q)$ are polynomial in $q$. Using results of Borel and Grushevsky-Hulek-Tommasi on cohomological stability, we adapt arguments of Achter-Erman-Kedlaya-Wood-Zureick-Brown to pose a conjecture about the asymptotics of the point counts $\#\mathcal{X}^{\times n}_g(\mathbb{F}_q)$ in the limit $g\rightarrow\infty$.Comment: 29 pages, comments welcome

NLR Genes Related Transcript Sets in Potato Cultivars Bearing Genetic Material of Wild Mexican Solanum Species

The long history of potato breeding includes the numerous introgressions of resistance genes from many wild species of South and Central America as well as from cultivated species into the breeding genepool. Most R genes belong to the NLR family with nucleotide-binding site–leucine-rich repeat. The aim of this research concerns an evaluation of NLR genes expression in transcriptomes of three potato cultivars (Evraziya, Siverskij, Sudarynya), which combine genetic material from wild and cultivated potato species, and each bears intragenic markers of RB/Rpi-blb1/Rpi-sto1 genes conferring broad-range resistance to late blight. The transcriptomes of the cultivars were compared before and 24 h after the Phytophthora infestans inoculation. The induction of RB/Rpi-blb1/Rpi-sto1 transcript after 24 h of inoculation was detected in the resistant cultivars Siverskij and Sudarynya but not in susceptible cv. Evraziya. This demonstrates the importance of transcriptomic assay for understanding the results of marker-assisted selection and phenotyping. Interestingly, assembling the transcriptomes de novo and analysis with NLR-parser tool revealed significant fractions of novel NLR genes with no homology to the reference genome from 103 (cv. Siverskij) to 160 (S. stoloniferum, 30514/15). Comparison of novel NLRs demonstrated a relatively small intersection between the genotypes that coincided with their complex pedigrees with several interspecific hybridization events. These novel NLRs may facilitate the discovery of new efficient R genes