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Register a new userDiagonal cubic forms and the large sieve
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Victor Y. Wang
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Let $F(boldsymbol{x})$ be a diagonal integer-coefficient cubic form in $min{4,5,6}$ variables. Excluding rational lines if $m=4$, we bound the number of integral solutions $boldsymbol{x}in[-X,X]^m$ to $F(boldsymbol{x})=0$ by $O_{F,epsilon}(X^{3m/4 - 3/2 + epsilon})$, conditionally on an optimal large sieve inequality (in a specific range of parameters) for approximate Hasse-Weil $L$-functions of smooth hyperplane sections $F(boldsymbol{x})=boldsymbol{c}cdotboldsymbol{x}=0$ as $boldsymbol{c}inmathbb{Z}^m$ varies in natural boxes. When $m$ is even, these results were previously established conditionally under Hooleys Hypothesis HW. Our $ell^2$ large sieve approach requires that certain bad factors be roughly $1$ on average in $ell^2$, while the $ell^infty$ Hypothesis HW approach only required the bound in $ell^1$. Furthermore, the large sieve only accepts uniform vectors; yet our initially given vectors are only approximately uniform over $boldsymbol{c}$, due to variation in bad factors and in the archimedean component. Nonetheless, after some bookkeeping, partial summation, and Cauchy, the large sieve will still apply. In an appendix, we suggest a framework for non-diagonal cubics, up to Hessian issues.
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