Rational curves on smooth cubic hypersurfaces over finite fields

Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension

Cubic hypersurfaces and a version of the circle method for number fields

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

Rapid computation of L-functions for modular forms

Let $f$ be a fixed (holomorphic or Maass) modular cusp form, with $L$-function $L(f,s)$. We describe an algorithm that computes the value $L(f,1/2+ iT)$ to any specified precision in time $O(1+|T|^{7/8})$

Rational points on complete intersections over ()

A two dimensional version of Farey dissection for function fields K = Fq(t) is developed and used to establish the quantitative arithmetic of the set of rational points on a smooth complete intersection of two quadrics X ⊂ P n−1 K , under the assumption that q is odd and n ≥ 9

Quartic forms in 30 variables

We will prove that smooth Quartic hypersurfaces satisfy the Hasse Principle as long as they are defined over at least 30 variables. The key tool here is employing Kloosterman's version of circle method. This is a joint work with Oscar Marmon (U Lund).Non UBCUnreviewedAuthor affiliation: Durham UniversityFacult

A sparse equidistribution result for (SL(2,R)/Γ0)n(\mathrm{SL}(2,\mathbb{R})/\Gamma_0)^{n}

Let G = SL(2, R) n, let Γ = Γn 0 , where Γ0 is a co-compact lattice in SL(2, R), let F(x) be a non-singular quadratic form and let u(x1, ..., xn) := 1 x1 0 1 ×...× 1 xn 0 1 denote unipotent elements in G which generate an n dimensional horospherical subgroup. We prove that in the absence of any local obstructions for F, given any x0 ∈ G/Γ, the sparse subset {u(x)x0 : x ∈ Z n, F(x) = 0} equidistributes in G/Γ as long as n ≥ 481, independent of the spectral gap of

An effective equidistribution result for SL(2,R) ∝(R2)⊕k and application to inhomogeneous quadratic forms

Let G=SL(2,R) (sic) (R-2)(circle plus k) and let Gamma be a congruence subgroup of SL(2,Z) (sic) (Z(2))(circle plus k). We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Gamma\G which project to pieces of closed horocycles in SL(2,Z)\SL(2,R). As an application, we prove an effective quantitative Oppenheim-type result for the quadratic form (m(1)-alpha)(2)+(m(2)-beta)(2)-(m(3)-alpha)(2)-(m(4)-beta)(2), for (alpha,beta)is an element of R-2 of Diophantine type, following the approach by Marklof [Ann. of Math. 158 (2003) 419-471] using theta sums