(2015) 'Rational points on cubic hypersurfaces over F(q,t).', Geometric and functional analysis., 25 (3). pp. 671-732. Further information on publisher's website:http://dx.doi.org/10.1007/s00039-015-0328-5Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s00039-015-0328-5Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. The Hasse principle and weak approximation is established for non-singular cubic hypersurfaces X over the function field F q (t), provided that char(F q ) > 3 and X has dimension at least 6.

The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Rational curves on smooth hypersurfaces of low degree Tim Browning and Pankaj VisheWe establish the dimension and irreducibility of the moduli space of rational curves (of fixed degree) on arbitrary smooth hypersurfaces of sufficiently low degree. A spreading out argument reduces the problem to hypersurfaces defined over finite fields of large cardinality, which can then be tackled using a function field version of the Hardy-Littlewood circle method, in which particular care is taken to ensure uniformity in the size of the underlying finite field.

We establish the Hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over Q.

Abstract. -A version of the Hardy-Littlewood circle method is developed for number fields K/Q and is used to show that non-singular projective cubic hypersurfaces over K always have a K-rational point when they have dimension at least 8.

Let M = Γ\ PSL(2, R) be a compact manifold, and let f ∈ C ∞ (M ) be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of f along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on M .