Rational lines on diagonal hypersurfaces and subconvexity via the circle method

Abstract

Fix $k,s,n\in \mathbb{N}$, and consider non-zero integers $c_1,\dots ,c_s$, not all of the same sign. Provided that $s\ge k(k+1)$, we establish a Hasse principle for the existence of lines having integral coordinates lying on the affine diagonal hypersurface defined by the equation $c_1{x}_1^k+\dots +c_s{x}_s^k=n$. This conclusion surmounts the conventional convexity barrier tantamount to the square-root cancellation limit for this problem.

22 pages