Research

For every n greater than or equal to 2, we determine the asymptotic formula for the number of integer triples (a,b,c) of bounded absolute value such that the generalised Fermat equation given by ax^n + by^n + cz^n = 0 is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations. 

The purpose of this article is twofold. On the one hand, we prove asymptotic formulas for the quantitative distribution of rational points on any smooth non-split projective quadratic surface. We obtain the optimal error term for the real place. On the other hand, we also study the growth of integral points on the three-dimensional punctured affine cone, as a quantitative version of strong approximation with Brauer--Manin obstruction for this quasi-affine variety. 

We prove asymptotics for semi-integral points of bounded height on toric varieties. We verify the Manin-type conjecture of Pieropan, Smeets, Tanimoto and Várilly-Alvarado for smooth and certain singular toric orbifolds upon replacing the leading constant with the one predicted by Chow, Loughran, Takloo-Bighash and Tanimoto. 

We derive asymptotic formulas for the number of rational points on a smooth projective quadratic hypersurface of dimension at least three inside of a shrinking adelic open neighbourhood. This is a quantitative version of weak approximation for quadrics and allows us to deduce the best growth rate of the size of such an adelic neighbourhood for which equidistribution is preserved. 

A central question in Arithmetic geometry is to determine for which polynomials f∈Z[t] and which number fields K the Hasse principle holds for the affine equation f(t)=N_{K/Q}(x)≠0. Whilst extensively studied in the literature, current results are largely limited to polynomials and number fields of low degree. In this paper, we establish the Hasse principle for a wide family of polynomials and number fields, including polynomials that are products of arbitrarily many linear, quadratic or cubic factors. The proof generalises an argument of Irving, which makes use of the beta sieve of Rosser and Iwaniec. As a further application of our sieve results, we prove new cases of a conjecture of Harpaz and Wittenberg on locally split values of polynomials over number fields, and discuss consequences for rational points in fibrations. 

We compare the Manin-type conjecture for Campana points recently formulated by Pieropan, Smeets, Tanimoto and Várilly-Alvarado with an alternative prediction of Browning and Van Valckenborgh in the special case of the orbifold  $(\mathbb{P}^1,D)$, where $D =\frac{1}{2}[0]+\frac{1}{2}[1]+\frac{1}{2}[\infty]$. We find that the two predicted leading constants do not agree, and we discuss whether thin sets could explain this discrepancy. Motivated by this, we provide a counterexample to the Manin-type conjecture for Campana points, by considering orbifolds corresponding to squareful values of binary quadratic forms. 

Poonen and Slavov recently developed a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing. In this paper, we extend their work by proving an analogous bound for the dimension of the exceptional locus in the setting of linear subspaces of higher codimensions. 

We find an asymptotic formula for the number of primitive vectors $(z_1,\ldots,z_4)\in (\mathbb{Z}_{\neq 0})^4$ such that $z_1,\ldots, z_4$ are all squareful and bounded by $B$, and $z_1+\cdots + z_4 = 0$. Our result agrees in the power of $B$ and $\log B$ with the Campana-Manin conjecture of Pieropan, Smeets, Tanimoto and V\'{a}rilly-Alvarado.