We study the number of intersections of the nodal lines of an eigenfunctionof the Laplacian on the standard torus with a fixed reference curve, that is,the number of zeros of the eigenfunction restricted to the curve. An upperbound is the wave number k. When the curve has nowhere zero curvature, weconjecture that, up to a constant multiple, this should also be the correct alower bound. We give a lower bound which differs from this by an arithmeticquantity, given in terms of the maximal number of lattice points in arcs ofsize square root of the wave number k on a circle of radius k. According to aconjecture of Cilleruelo and Granville, this quantity is bounded in which casewe recover our conjecture. To get at the lower bound, we reduce the problem togiving a lower bound for the L1 norm of the restriction of the eigenfunction tothe curve, and then to an upper bound for the L4 restriction norm.
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