The analytic rank of a family of jacobians of Fermat curves

• Autorzy: Jędrzejak T.
• Języki publikacji: EN

Abstrakty

EN

We study the family of curves Fm(p) : xp + yp = m, where p is an odd prime and m is a pth power free integer. We prove some results about the distribution of root numbers of the L-functions of the hyperelliptic curves associated to the curves Fm(p). As a corollary we conclude that the Jacobians of the curves Fm(5) with even analytic rank and those with odd analytic rank are equally distributed.

Słowa kluczowe

EN

twisted Fermat curves; hyperelliptic curves; jacobian variety; L-function; root number; analytic rank

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2008
• Tom: Vol. 56, no 3-4
• Strony: 199--206
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Jędrzejak T.

• Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland,

Bibliografia

• [1] A. Dąbrowski and T. Jędrzejak, Ranks in families of Jacobian varieties of twisted Fermat curves, Canad. Math. Bull., to appear.
• [2] A. Dąbrowski and J. Pomykała, Nonvanishing of motivic L-functions, Math. Proc. Cambridge Philos. Soc. 130 (2001), 221-235.
• [3] D. K. Faddeev, The group of divisor classes on some algebraic curves, Soviet Math. Dokl. 2 (1961), 67-69.
• [4] —, Invariants of divisor classes for the curves xk(l — x) = yl in the l-adic cyclotomic field, Trudy Mat. Inst. Steklov. 64 (1961), 284-293 (in Russian).
• [5] H. Iwaniec and P. Sarnak, The non-vanishing of central values of automorphic L-functions and Landau-Siegel zeros, Israel J. Math. 120 (2000), 155-177.
• [6] L. Mai, The analytic rank of a family of elliptic curves, Canad. J. Math. 45 (1993), 847-862.
• [7] W. Narkiewicz, Number Theory, Biblioteka Mat. 50, PWN, Warszawa, 1977 (in Polish); English transl.: World Sci., 1983.
• [8] M. Stoll, On the arithmetic of the curves y2 = xl + A, II, J. Number Theory 93 (2002), 183-206.