On the number of linear spaces on hypersurfaces with a prescribed discriminant

Abstract

For a given form F ∈ Z[x 1 , . . . , x s ] we apply the circle method in order to give an asymptotic estimate of the number of m-tuples x 1 , . . . , x m spanning a linear space on the hypersurface F(X ) = 0 with the property that det ((x 1 , . . . , x m ) t (x 1 , . . . , x m )) = b. This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.

Linear Spaces on Hypersurfaces over Number Fields

“…The constant c has an interpretation as a product of local densities, so that Theorem 1.1 yields an analytic Hasse principle. We also note that the case m = 1 recovers Skinner's result [26], and for larger m we save approximately one factor r over what a naive application of Skinner's methods would yield, thus replicating the improvements of the author's earlier work [2,4] over a naive application of Birch's theorem. One feature of the proof that is worth highlighting is our treatment of the singular integral.…”

“…We set up the circle method as in [2]. The additive character over number fields is given by e(x) = e 2πi Tr x .…”

“…Similarly, the recent paper of Browning and Heath-Brown generalising Birch's theorem to systems involving differing degrees [7] has immediately been translated to the number field setting by Frei and Madritsch [12], as has Dietmann's work on small solutions of quadratic forms [11] by Helfrich [16]. In this memoir we aim to continue in this direction by providing a number field version of the author's recent work on linear spaces on hypersurfaces [2,4].…”

The density of rational lines on hypersurfaces: a bihomogeneous perspective

“…When F is a cubic form, recent work of the author jointly with Dietmann [12] shows that (1.1) has non-trivial rational solutions whenever n ≥ 29, but that there may not be any rational solutions when n = 11 or lower. For more general settings, (1.1) has been investigated in a series of papers by the present author [8][9][10][11]. We note at this point that, in order to strictly count lines, we would have to exclude those solutions of (1.1) where x and y are proportional.…”

The density of rational lines on hypersurfaces: A bihomogeneous perspective

“…When F is a cubic form, recent work of the author jointly with Dietmann [6] shows that the equation (1.1) has non-trivial rational solutions whenever n 29, but that there may not be any rational solutions when n = 11 or lower. For more general settings, the equation (1.1) has been investigated in a series of papers by the present author [2,3,4,5]. We note at this point that, in order to strictly count lines, we would have to exclude those solutions of (1.1) where x and y are proportional.…”