LEADER 03673nam 2200697 450 001 9910813546003321 005 20201204082343.0 010 $a1-4704-6253-2 035 $a(CKB)4100000011437433 035 $a(MiAaPQ)EBC6346635 035 $a(RPAM)21697713 035 $a(PPN)250799235 035 $a(EXLCZ)994100000011437433 100 $a20201204d2020 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aExplicit arithmetic of Jacobians of generalized Legendre curves over global function fields /$fLisa Berger [and seven others] 210 1$aProvidence, Rhode Island :$cAmerican Mathematical Society,$d[2020] 210 4$dİ2020 215 $a1 online resource (144 pages) 225 1 $aMemoirs of the American Mathematical Society ;$vNumber 1295 300 $a"Forthcoming, volume 266, number 1295." 311 $a1-4704-4219-1 320 $aIncludes bibliographical references. 327 $aThe curve, explicit divisors, and relations -- Descent calculations -- Minimal regular model, local invariants, and domination by a product of curves -- Heights and the visible subgroup -- The L-function and the BSD conjecture -- Analysis of J[p] and NS(Xd)tor -- Index of the visible subgroup and the Tate-Shafarevich group -- Monodromy of ?-torsion and decomposition of the Jacobian. 330 $a"We study the Jacobian J of the smooth projective curve C of genus r-1 with affine model yr = xr-1(x+ 1)(x + t) over the function field Fp(t), when p is prime and r [greater than or equal to] 2 is an integer prime to p. When q is a power of p and d is a positive integer, we compute the L-function of J over Fq(t1/d) and show that the Birch and Swinnerton-Dyer conjecture holds for J over Fq(t1/d). When d is divisible by r and of the form p[nu] + 1, and Kd := Fp([mu]d, t1/d), we write down explicit points in J(Kd), show that they generate a subgroup V of rank (r-1)(d-2) whose index in J(Kd) is finite and a power of p, and show that the order of the Tate-Shafarevich group of J over Kd is [J(Kd) : V ]2. When r > 2, we prove that the "new" part of J is isogenous over Fp(t) to the square of a simple abelian variety of dimension [phi](r)/2 with endomorphism algebra Z[[mu]r]+. For a prime with pr, we prove that J[](L) = {0} for any abelian extension L of Fp(t)"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society ;$vNumber 1295. 606 $aCurves, Algebraic 606 $aLegendre's functions 606 $aRational points (Geometry) 606 $aBirch-Swinnerton-Dyer conjecture 606 $aJacobians 606 $aAbelian varieties 606 $aFinite fields (Algebra) 615 0$aCurves, Algebraic. 615 0$aLegendre's functions. 615 0$aRational points (Geometry) 615 0$aBirch-Swinnerton-Dyer conjecture. 615 0$aJacobians. 615 0$aAbelian varieties. 615 0$aFinite fields (Algebra) 676 $a516.352 686 $a11G10$a11G30$a11G40$a14G05$a14G25$a14K15$2msc 700 $aBerger$b Lisa$f1969-$01686950 702 $aHall$b Chris$f1975- 702 $aPannekoek$b Rene? 702 $aPark$b Jennifer Mun Young 702 $aPries$b Rachel$f1972- 702 $aSharif$b Shahed$f1977- 702 $aSilverberg$b Alice 702 $aUlmer$b Douglas$f1960- 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910813546003321 996 $aExplicit arithmetic of Jacobians of generalized Legendre curves over global function fields$94060057 997 $aUNINA