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We compute the Mordell–Weil groups of the modular Jacobian varieties of hyperelliptic modular curves X 1 (M , M N) over every composite field of some quadratic number fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman. [ABSTRACT FROM AUTHOR]

The cerebellum is involved in some forms of motor coordination and motor learning. Here we isolated transgenic (Tg) zebrafish lines that express a modified version of Gal4-VP16 (GFF) in the cerebellar neural circuits: granule, Purkinje, or eurydendroid cells, Bergmann glia, or the neurons in the inferior olive nuclei (IO) which send climbing fibers to Purkinje cells, with the transposon Tol2 system. By combining GFF lines with Tg lines carrying a reporter gene located downstream of Gal4 binding sequences (upstream activating sequence: UAS), we investigated the anatomy and developmental processes of the cerebellar neural circuitry. Combining an IO-specific Gal4 line with a UAS reporter line expressing the photoconvertible fluorescent protein Kaede demonstrated the contralateral projections of climbing fibers. Combining a granule cell-specific Gal4 line with a UAS reporter line expressing wheat germ agglutinin (WGA) confirmed direct and/or indirect connections of granule cells with Purkinje cells, eurydendroid cells, and IO neurons in zebrafish. Time-lapse analysis of a granule cell-specific Gal4 line revealed initial random movements and ventral migration of granule cell nuclei. Transgenesis of a reporter gene with another transposon Tol1 system visualized neuronal structure at a single cell resolution. Our findings indicate the usefulness of these zebrafish Gal4 Tg lines for studying the development and function of cerebellar neural circuits. [ABSTRACT FROM AUTHOR]