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ALGEBRAIC VARIETIES --- BIRCH-SWINNERTON-DYER CONJECTURE --- KEPLER'S CONJECTURE --- MATHEMATICS --- MATHEMATICS --- TOPOLOGY --- TURBULENCE --- HISTORY --- PHILOSOPHY --- ALGEBRAIC VARIETIES --- BIRCH-SWINNERTON-DYER CONJECTURE --- KEPLER'S CONJECTURE --- MATHEMATICS --- MATHEMATICS --- TOPOLOGY --- TURBULENCE --- HISTORY --- PHILOSOPHY

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"We study the Jacobian J of the smooth projective curve C of genus r-1 with affine model yr = xr-1(x+ 1)(x + t) over the function field Fp(t), when p is prime and r [greater than or equal to] 2 is an integer prime to p. When q is a power of p and d is a positive integer, we compute the L-function of J over Fq(t1/d) and show that the Birch and Swinnerton-Dyer conjecture holds for J over Fq(t1/d). When d is divisible by r and of the form p[nu] + 1, and Kd := Fp([mu]d, t1/d), we write down explicit points in J(Kd), show that they generate a subgroup V of rank (r-1)(d-2) whose index in J(Kd) is finite and a power of p, and show that the order of the Tate-Shafarevich group of J over Kd is [J(Kd) : V ]2. When r > 2, we prove that the "new" part of J is isogenous over Fp(t) to the square of a simple abelian variety of dimension [phi](r)/2 with endomorphism algebra Z[[mu]r]+. For a prime with pr, we prove that J[](L) = {0} for any abelian extension L of Fp(t)"--

Torsion --- Torsion (mécanique) --- Algebraic functions --- Fonctions algébriques --- Nombres, Théorie des --- Number theory --- Curves, Algebraic. --- Abelian varieties. --- Jacobians. --- Birch-Swinnerton-Dyer conjecture. --- Rational points (Geometry) --- Legendre's functions. --- Finite fields (Algebra)