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All Outputs (11)

On the Hasse Principle For Complete Intersections (2024)
Journal Article
Northey, M. J., & Vishe, P. (2024). On the Hasse Principle For Complete Intersections. Compositio Mathematica, 160(4), 771-835. https://doi.org/10.1112/S0010437X23007698

We prove the Hasse principle for a smooth projective variety X ⊂ P n−1 Q defined by a system of two cubic forms F, G as long as n ≥ 39. The main tool here is the development of a version of Kloosterman refinement for a smooth system of equations defi... Read More about On the Hasse Principle For Complete Intersections.

Rational points on complete intersections over 𝔽𝒒(𝒕) (2022)
Journal Article
Vishe, P. (2023). Rational points on complete intersections over 𝔽𝒒(𝒕). Proceedings of the London Mathematical Society, 126(2), 556-619. https://doi.org/10.1112/plms.12496

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 assumpt... Read More about Rational points on complete intersections over 𝔽𝒒(𝒕).

A sparse equidistribution result for $(\mathrm{SL}(2,\mathbb{R})/\Gamma_0)^{n}$ (2021)
Journal Article
Vishe, P. (2022). A sparse equidistribution result for $(\mathrm{SL}(2,\mathbb{R})/\Gamma_0)^{n}$. Transactions of the American Mathematical Society, 375(1), 669-694. https://doi.org/10.1090/tran/8498

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... Read More about A sparse equidistribution result for $(\mathrm{SL}(2,\mathbb{R})/\Gamma_0)^{n}$.

An effective equidistribution result for SL(2,R)\ltimes (R^2)^{oplus k} and application to inhomogeneous quadratic forms (2020)
Journal Article
Strombergsson, A., & Vishe, P. (2020). An effective equidistribution result for SL(2,R)\ltimes (R^2)^{oplus k} and application to inhomogeneous quadratic forms. Journal of the London Mathematical Society, 102(1), 143-204. https://doi.org/10.1112/jlms.12316

Let G = SL(2, R) (R2) ⊕k and let Γ be a congruence subgroup of SL(2, Z) (Z2) ⊕k. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in Γ\G which project to pieces of closed horocycles in SL(2, Z... Read More about An effective equidistribution result for SL(2,R)\ltimes (R^2)^{oplus k} and application to inhomogeneous quadratic forms.

Diophantine approximation for products of linear maps—logarithmic improvements (2016)
Journal Article
Gorodnik, A., & Vishe, P. (2016). Diophantine approximation for products of linear maps—logarithmic improvements. Transactions of the American Mathematical Society, 370(1), 487-507. https://doi.org/10.1090/tran/6953

This paper is devoted to the study of a problem of Cassels in multiplicative Diophantine approximation which involves minimising values of a product of affine linear forms computed at integral points. It was previously known that values of this produ... Read More about Diophantine approximation for products of linear maps—logarithmic improvements.

Uniform Bounds for Period Integrals and Sparse Equidistribution (2015)
Journal Article
Tanis, J., & Vishe, P. (2015). Uniform Bounds for Period Integrals and Sparse Equidistribution. International Mathematics Research Notices, 2015(24), 13728-13756. https://doi.org/10.1093/imrn/rnv115

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 o... Read More about Uniform Bounds for Period Integrals and Sparse Equidistribution.

Cubic hypersurfaces and a version of the circle method for number fields (2014)
Journal Article
Browning, T., & Vishe, P. (2014). Cubic hypersurfaces and a version of the circle method for number fields. Duke Mathematical Journal, 163(10), 1825-1883. https://doi.org/10.1215/00127094-2738530

A version of the Hardy–Littlewood circle method is developed for number fields K/QK/Q and is used to show that nonsingular projective cubic hypersurfaces over KK always have a KK-rational point when they have dimension at least 88.