An introduction to the determinant method: Part III

In this latest installment on the determinant method (see here and here for the previous two posts), I will finally discuss something I have actually completed work on: the $p$-adic determinant method in the setting of weighted projective spaces.

In the original formulation of the determinant method due to Heath-Brown, he showed that the quality of the estimates involving the determinant method heavily depends on the dimension of the variety. In particular, the factor that one uses to control the number of auxiliary polynomials needed depends on the factor $1/d^{1/n}$, where $n$ is the dimension of the variety. Thus, the larger $n$ gets, the closer this gets to unity if the degree is fixed. Therefore, one can hope to do better by considering a variety defined by homogeneous polynomials as a projective variety, since then the dimension gets reduced by one. Aside from this, Heath-Brown did not appear to use any other notions from algebraic geometry, and his proof in his 2002 Annals paper used ad-hoc arguments that also did not seem to relate much to algebraic geometry. However, Broberg soon formulated Heath-Brown’s theorem in terms of algebraic geometry and thus significantly solidified the theoretical backbone of Heath-Brown’s machinery. Consequently, Salberger established Heath-Brown’s theorem in the language of algebraic geometry. In this setting, it became clear how to generalize Heath-Brown’s theorem to other types of projective spaces, in particular weighted projective spaces.

The advantage is that we are able to deal with varieties with good algebraic structure, but for which the natural counting function for heights of bounded height involve very skew boxes. The standard determinant method for projective spaces work best with boxes that are roughly cubes. A prototypical example is the following two problems:

Question 1: Let $f(x_1, x_2, x_3)$ be a ternary form with even degree. Count the number of rational points on the variety defined by $f(x_1, x_2, x_3) = t^2.$

Question 2: Let $f(x,y)$ be a binary form of degree $d \geq 3$ with integral coefficients, which is not too degenerate. Estimate the density of pairs of integers $(x,y)$ such that $f(x,y)$ is square-free.

In the first question, it is natural to define the counting function as $N_f(B) = \# \{(x_1, x_2, x_3 \in \mathbb{Z}^3 : \gcd(x_1, x_2, x_3) = 1, |x_i| \leq B\}$. Making this choice, we realize that $|t|$ could be as large as $B^{d/2}$. Thus, we would be considering very lopsided boxes. In the second question, after some non-trivial work due to George Greaves, one can show that the question is reduced to counting solutions to the equation $f(x,y) = vz^2$

where $|x|, |y| \leq B$, $|z| \ll B^\beta$ where $2 < \beta < d/2$, and $|v| \ll B^{d - 2 \beta}$ (see my paper).

The first type of variety should be naturally considered to be a surface in the weighted projective space $\mathbb{P}(1,1,1,d/2)$, and the latter should be considered as a surface in $\mathbb{P}(1,1,d-4, 2)$.

The advantage of considering these as surfaces in a suitable weighted projective space is that we can now take advantage of the structure of weighted projective spaces to `equalize’ the box size, so that the large outlier bounds are properly taken care of. The trade-off is that the $1/d^{1/r}$ term in the exponent of the determinant method is replaced with $(w/d)^{1/r}$, where $w$ is the product of all of the weights and $r$ is the dimension of the variety. This trade-off in fact produces superior results, as opposed to considering the variety as a hypersurface in $\mathbb{A}^4$, say.

The way to approach the determinant method in the weighted projective setting is to obtain the correct analogues of the Hilbert function. This has been worked out by Dolgachev in a seminal 1982 paper, where many of the basic notions of weighted projective spaces are worked out. Having established the correct algebraic geometric machinery for weighted projective spaces, it is then a relatively simple matter to translate the work of Salberger into the new setting. This is the focus of this paper.