Class invariants for quartic CM fields

One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S-units in certain abelian extensions of K, where S is effectively determined by K. It also yields class polynomials for primitive quartic CM fields whose coefficients are S-integers.Comment: 14 page

The Humanistic Mathematics Network Journal: A Bibliographic Report

The content of the Humanistic Mathematics Network Newsletter was reviewed by Claire Skrivanos and Qingcheng Zhang in [1]. This report reviews the content of the Humanistic Mathematics Network Journal (1992-2004)

In a more polarized era more and more citizens are structuringtheir beliefs along ideological lines, just as politicians do

While politics in the US has long been split along the left-right, liberal-conservative spectrum, most academics have assumed that these divisions had historically not percolated through to how most Americans see themselves. In new research which draws on election data, Caitlin E. Jewitt and Paul N. Goren find that people are now more able to structure their beliefs along the left-right dimension than they were in 1980. As debates between political elites have become more polarized, many more Americans who are ideologically engaged are now thinking about issues and organizing their policy positions just as politicians do

Ichthyological Bulletin of the JLB Smith Institute of Ichthyology; No. 58

The following 84 gobioid fishes are reported from the Maldive Islands (those preceded by asterisk represent new records). GOBIIDAE: Amblyeleotris aurora (Polunin & Lubbock), *A. diagonalis Polunin and Lubbock, *A. periophthalma (Bleeker), *A. steinitzi (Klausewitz), A. wheeleri (Polunin & Lubbock), * Amblygobius hectori (Smith), A. semicinctus (Bennett), * Asterropteryx semipunctatus Ruppell, *A. spinosus (Goren), *Bathygobius calitus (Bennett), B. cocosensis (Bleeker), *B. cyclopterus (Valenciennes), * Cabillus tongarevae (Fowler), * Callogobius centrolepis Weber, *C. sclateri (Steindachner), *C. sp., Cryptocentrus fasciatus (Playfair & Gunther), *Ctenogobiops crocineus Smith, C. feroculus Lubbock & Polunin, * Eviota albolineata Jewett & Lachner, *E. guttata Lachner & Karnella, *E. nebulosa Smith, *E. nigripinna Lachner & Karnella, *E. prasina (Kluzinger), *E. sebreei Jordan & Seale, *E. zebrina Lachner & Karnella, *E. sp., * Flabelligobius latruncularius (Klausewitz), * Fusigobius duospilus Hoese & Reader, *F. neophytus (Gunther), *F. sp. 1 (sp. A of Winterpottom & Emery, 1986), *F. sp. 2 (sp. B of Winterbottom & Emery, 1986), *Gnatholepis anjerensis (Bleeker), *G. scapulostigma Herre, *Gobiodon citrinus (Ruppell), *G. sp. (Chagos specimens identified as G. rivulatus by Winterbottom & Emery, 1986), *Hetereleotris zanzibarensis (Smith), *Istigobius decoratus (Herre), *Macrodontogobius wilburi Herre, Oplopomus caninoides (Bleeker) (reported from Maldives by Regan, 1908), O. Oplopomus (Valenciennes) (reported from Maldives by Regan, 1908, as Hoplopomus acanthistius), *Opua maculipinnis, n. sp. (Opua E.K. Jordan is regarded as a senior synonym of Oplopomops Smith; the new species is characterized as follows: no dorsal spines filamentous, the third longest; 10 soft rays in second dorsal and anal fins; 27 scales in longitudinal series on body, 9 prodorsal scales; body depth 4.9 in SL, a midlateral row of five dusky blotches on body each containing a pair of dark brown spots, a large dusky spot under eye, and a large black spot posteriorly in first dorsal fin), *Palutrus reticularis Smith,* Papillogobius reichei (Bleeker), *Paragobiodon lacuniculus (Kendall and Goldsborough), *P. modestus (Regan), *Pleurosicya michelli Fourmanoir, *Priolepis cinctus (Regan), *P. nocturnus (Smith), *P. semidoliatus (Valenciennes), P. sp., Stonogobiops dracula Lubbock & Polunin, * Sueviota lachneri Winterbottom & Hoese, *Trimma emeryi Winterbottom, *T. flammeum (Smith), *T. naudei Smith, *T. striata (Herre), *T. taylori Lobel, *T. tevegae Cohen & Davis, *T sp. 1, *T. sp. 2, *T. sp. 3, *T. sp. 4 (these four species of trimma to be described by R. Winterbottom), *Trimmatom nanus Winterbottom & Emery, Valenciennea helsdingenii (Bleeker), V. puellaris (Tomiyama), V. sexguttata (Valenciennes), V. strigata (Broussonet), V. sp. (to be named by Hoese and Larson, in press), Vanderhorstia ambanoro (Fourmanoir),* V. ornatissima Smith, V. prealta Lachner & McKinney. ELEOTRIDIDAE: Eleotris melanosoma Bleeker. MICRODESMIDAE: * Gunnellichthys curiosus Dawson, *G. monostigma Smith, G. viridescens Dawson, *Ne- mateleotris decora Randall & Allen, N. magnifica Fowler, Ptereleotris evides (Jordan & Hubbs), P. heteroptera Bleeker, *P. microlepis (Bleeker), *P. zebra (Fowler), *P. sp. (probably either P. hanae or P. arabica; specimen needed). XENISTHMIDAE: Xenisthmus polyzonatus (Klunzinger).Digitised by Rhodes University Library on behalf of SAIA

A Gross-Zagier formula for quaternion algebras over totally real fields

We prove a higher dimensional generalization of Gross and Zagier's theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields $K$ and $K^\prime$, when the curves are reduced modulo a supersingular prime and its powers. Equivalently, the Gross-Zagier formula counts optimal embeddings of the ring of integers of an imaginary quadratic field into particular maximal orders in $B_{p, \infty}$, the definite quaternion algebra over \QQ ramified only at $p$ and infinity. Our work gives an analogous counting formula for the number of simultaneous embeddings of the rings of integers of primitive CM fields into superspecial orders in definite quaternion algebras over totally real fields of strict class number 1. Our results can also be viewed as a counting formula for the number of isomorphisms modulo $\frak{p} | p$ between abelian varieties with CM by different fields. Our counting formula can also be used to determine which superspecial primes appear in the factorizations of differences of values of Siegel modular functions at CM points associated to two different CM fields, and to give a bound on those supersingular primes which can appear. In the special case of Jacobians of genus 2 curves, this provides information about the factorizations of numerators of Igusa invariants, and so is also relevant to the problem of constructing genus 2 curves for use in cryptography.Comment: 32 page