New methods for bounding the number of points on curves over finite fields

  • Everett W. Howe

    Center for Communications Research, San Diego, USA
  • Kristin E. Lauter

    Theory Group, Redmond, USA
New methods for bounding the number of points on curves over finite fields cover
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Abstract

We provide new upper bounds on Nq(g)N_q(g), the maximum number of rational points on a smooth absolutely irreducible genus-gg curve over Fq\mathbb {F}_q, for many values of qq and gg. Among other results, we find that N4(7)=21N_4(7) = 21 and N8(5)=29N_8(5) = 29, and we show that a genus-12 curve over F2\mathbb {F}_2 having 15 rational points must have characteristic polynomial of Frobenius equal to one of three explicitly given possibilities.

We also provide sharp upper bounds for the lengths of the shortest vectors in Hermitian lattices of small rank and determinant over the maximal orders of small imaginary quadratic fields of class number 1.

Some of our intermediate results can be interpreted in terms of Mordell–Weil lattices of constant elliptic curves over one-dimensional function fields over finite fields. Using the Birch and Swinnerton-Dyer conjecture for such elliptic curves, we deduce lower bounds on the orders of certain Shafarevich–Tate groups.