The a-number of hyperelliptic curves

It is known that for a smooth hyperelliptic curve to have a large $a$-number, the genus must be small relative to the characteristic of the field, $p>0$, over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve $C$ to have $a_C=g-1$, the genus is bounded by $g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g <p$. The method of proof is to force the Cartier-Manin matrix to have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. We then use this bound to summarize what is known about the existence of such curves when $p=3,5$ and $7$.Comment: 7 pages. v2: revised and improved the proof of the main theorem based on suggestions from the referee. To appear in the proceedings volume of Women in Numbers Europe-

A human colonic crypt culture system to study regulation of stem cell-driven tissue renewal and physiological function

The intestinal epithelium is one of the most rapidly renewing tissues in the human body and fulfils vital physiological roles such as barrier function and transport of nutrients and fluid. Investigation of gut epithelial physiology in health and disease has been hampered by the lack of ex vivo models of the native human intestinal epithelium. Recently, remarkable progress has been made in defining intestinal stem cells and in generating intestinal organoid cultures. In parallel, we have developed a 3D culture system of the native human colonic epithelium that recapitulates the topological hierarchy of stem cell-driven tissue renewal and permits the physiological study of native polarized epithelial cells. Here we describe methods to establish 3D cultures of intact human colonic crypts and conduct real-time imaging of intestinal tissue renewal, cellular signalling, and physiological function, in conjunction with manipulation of gene expression by lentiviral or adenoviral transduction. Visualization of mRNA- and protein-expression patterns in cultured human colonic crypts, and cross-validation with crypts derived from fixed mucosal biopsies, is also described. Alongside studies using intestinal organoids, the near-native human colonic crypt culture model will help to bridge the gap that exists between investigation of colon cancer cell lines and/or animal (tissue) studies, and progression to clinical trials. To this end, the near native human colonic crypt model provides a platform to aid the development of novel strategies for the prevention of inflammatory bowel disease and cancer

Automorphic Instanton Partition Functions on Calabi-Yau Threefolds

We survey recent results on quantum corrections to the hypermultiplet moduli space M in type IIA/B string theory on a compact Calabi-Yau threefold X, or, equivalently, the vector multiplet moduli space in type IIB/A on X x S^1. Our main focus lies on the problem of resumming the infinite series of D-brane and NS5-brane instantons, using the mathematical machinery of automorphic forms. We review the proposal that whenever the low-energy theory in D=3 exhibits an arithmetic "U-duality" symmetry G(Z) the total instanton partition function arises from a certain unitary automorphic representation of G, whose Fourier coefficients reproduce the BPS-degeneracies. For D=4, N=2 theories on R^3 x S^1 we argue that the relevant automorphic representation falls in the quaternionic discrete series of G, and that the partition function can be realized as a holomorphic section on the twistor space Z over M. We also offer some comments on the close relation with N=2 wall crossing formulae.Comment: 25 pages, contribution to the proceedings of the workshop "Algebra, Geometry and Mathematical Physics", Tjarno, Sweden, 25-30 October, 201

Elliptic Calabi-Yau threefolds over a del Pezzo surface

We consider certain elliptic threefolds over the projective plane (more generally over certain rational surfaces) with a section in Weierstrass normal form. In particular, over a del Pezzo surface of degree 8, these elliptic threefolds are Calabi–Yau threefolds. We will discuss especially the generating functions of Gromov–Witten and Gopakumar–Vafa invariants

Experimental evidence for core-Merge in the vocal communication system of a wild passerine

シジュウカラに言語の核：2語を1つにまとめる力（併合）を確認. 京都大学プレスリリース. 2022-09-29.One of the cognitive capacities underlying language is core-Merge, which allows senders to combine two words into a sequence and receivers to recognize it as a single unit. Recent field studies suggest intriguing parallels in non-human animals, e.g., Japanese tits (Parus minor) combine two meaning-bearing calls into a sequence when prompting antipredator displays in other individuals. However, whether such examples represent core-Merge remains unclear; receivers may perceive a two-call sequence as two individual calls that are arbitrarily produced in close time proximity, not as a single unit. If an animal species has evolved core-Merge, its receivers should treat a two-call sequence produced by a single individual differently from the same two calls produced by two individuals with the same timing. Here, we show that Japanese tit receivers exhibit antipredator displays when perceiving two-call sequences broadcast from a single source, but not from two sources, providing evidence for core-Merge in animals

Ancient evolutionary origin of vertebrate enteric neurons from trunk-derived neural crest

The enteric nervous system of jawed vertebrates arises primarily from vagal neural crest cells that migrate to the foregut and subsequently colonize and innervate the entire gastrointestinal tract. Here we examine development of the enteric nervous system in the basal jawless vertebrate the sea lamprey (Petromyzon marinus) to gain insight into its evolutionary origin. Surprisingly, we find no evidence for the existence of a vagally derived enteric neural crest population in the lamprey. Rather, labelling with the lipophilic dye DiI shows that late-migrating cells, originating from the trunk neural tube and associated with nerve fibres, differentiate into neurons within the gut wall and typhlosole. We propose that these trunk-derived neural crest cells may be homologous to Schwann cell precursors, recently shown in mammalian embryos to populate post-embryonic parasympathetic ganglia, including enteric ganglia. Our results suggest that neural-crest-derived Schwann cell precursors made an important contribution to the ancient enteric nervous system of early jawless vertebrates, a role that was largely subsumed by vagal neural crest cells in early gnathostomes

Counting points on hyperelliptic curves over finite fields

International audienceWe describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature