We prove the existence of designs of small size in a number of contexts. In particular our techniques can be applied to prove the existence of n-designs on Sd of size Od(nd logd−1(n)).

AbstractWe study the variation of the φ-Selmer groups of the elliptic curvesy2=x3−Dxunder quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integerDis constrained to lie in a family for which the relative Tamagawa number of the isogeny φ is fixed.