Equidistribution of values of linear forms on a cubic hypersurface

Abstract

Let $C$ be a cubic form with rational coefficients in $n$ variables, and let $h$ be the $h$-invariant of $C$. Let $L_1,\dots ,L_r$ be linear forms with real coefficients such that if $\mathbf{\alpha }\in {\mathbb{R}}^r\setminus \{0\}$ then $\mathbf{\alpha }\cdot \mathbf{L}$ is not a rational form. Assume that $h>16+8r$. Let $\mathbf{\tau }\in {\mathbb{R}}^r$, and let $\eta$ be a positive real number. We prove an asymptotic formula for the weighted number of integer solutions $\mathbf{x}\in [-P,P{]}^n$ to the system $C(\mathbf{x})=0,|\mathbf{L}(\mathbf{x})-\mathbf{\tau }|<\eta$. If the coefficients of the linear forms are algebraically independent over the rationals, then we may replace the $h$-invariant condition with the hypothesis $n>16+9r$, and show that the system has an integer solution. Finally, we show that the values of $\mathbf{L}$ at integer zeros of $C$ are equidistributed modulo one in ${\mathbb{R}}^r$, requiring only that $h>16$.

References in corpus (1)

• Rational points on intersections of cubic and quadric hypersurfaces