Rational lines on diagonal hypersurfaces and subconvexity via the circle method

Abstract

Fix k , s , n ∈ N k,s,n\in \mathbb {N} , and consider non-zero integers c 1 , … , c s c_1,\ldots ,c_s , not all of the same sign. Provided that s ⩾ k ( k + 1 ) s\geqslant 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 + … + c s x s k = n c_1x_1^k+\ldots +c_sx_s^k=n . This conclusion surmounts the conventional convexity barrier tantamount to the square-root cancellation limit for this problem.