Improved bounds in Birch's theorem for forms in many variables
We improve the best known result on the number of variables needed for the smooth Hasse principle for homogeneous forms of degree \(d\geq5\).
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arXiv subjects
Yijie Diao
Publications and source records attributed to Yijie Diao.
Average root numbers in two isotrivial families of elliptic curves
We study root numbers in the isotrivial families $y^2=x^3+a$ and $y^2=x^3+ax$. For a broad class of fixed binary forms, we prove that the average root number over primitive pairs exists. Assuming finiteness of the relevant Tate--Shafarevich groups, we deduce Zariski density for certain del Pezzo surfaces of degree $1$ arising from separable binary sextics. We also establish explicit averages of root numbers for almost all polynomials of each fixed degree $d\geq2$, ordered by coefficient height. The proof develops a new quantitative transference principle for polynomial values.
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Liouville function, von Mangoldt function and norm forms at random binary forms
We analyze the average behavior of various arithmetic functions at the values of degree $d$ binary forms ordered by height, with probability $1$. This approach yields averaged versions of the Chowla conjecture and the Bateman-Horn conjecture for random binary forms. Furthermore, we show that the rational Hasse principle holds for almost all Châtelet varieties defined by a fixed norm form of degree $e$ and by varying binary forms of fixed degree $d$, provided $e$ divides $d$. This proves an average version of a conjecture of Colliot-Thélène.
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Class numbers and integer points on some Pellian surfaces
We provide an estimate for the number of nontrivial integer points on the Pellian surface $t^2 - du^2 = 1$ in a bounded region. We give a lower bound on the size of fundamental solutions for almost all $d$ in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.
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Density of the union of positive diagonal binary quadratic forms
Let $X$ be a sufficiently large positive integer. We prove that one may choose a subset $S$ of primes with cardinality $O(\log X)$, such that a positive proportion of integers less than $X$ can be represented by $x^2 + p y^2$ for at least one of $p \in S$.
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