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Jedrzejak, Tomasz
- Partitions of Graphs and Selmer Groups of Elliptic Curves of Neumann-Setzer Type
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1 University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, PL
1 University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, PL
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Journal of the Ramanujan Mathematical Society, Vol 29, No 1 (2014), Pagination: 75-92Abstract
We consider the elliptic curves Eu : y2 = x3 + ux2 − 16x and their quadratic twists Eun by a squarefree integer n, where u2 + 64 = p1 . . . pl, (pi are primes). When l ≤ 2, n ≡ 1(mod 4) and all prime divisors of n are congruent to 3 modulo 4 we give a complete description of sizes of Selmer groups of Eun in terms of number of even partitions of some graphs. If n is even or l > 2, we give some conditions for twists of rank zero. We deduce also that Eun has rank zero for a positive proportion of squarefree integers n with a fixed number of prime divisors.
- Polynomial Pell Equations P(x)2 − (x2m + Ax + B)q(x)2 = 1 and Associated Hyperelliptic Curves
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1 University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, PL
1 University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, PL
Source
Journal of the Ramanujan Mathematical Society, Vol 34, No 2 (2019), Pagination: 263-269Abstract
The title equations are connected with Jacobians of hyperelliptic curves Cm,a,b : y2 = x2m + ax + b defined over Q. More precisely, these equations have a nontrivial solution if and only if the class of the divisor ∞+ − ∞− is a torsion point in Jacobian Jac(Cm,a,b), where ∞+ and ∞− are two points at infinity in Cm,a,b. We show that if ab = 0 then the title equations have nontrivial solutions (and we write explicit formulae). On the other hand, we prove that for any m > 1 there exist infinitely many pairs (a, b) such that our equations have no nontrivial solutions. Moreover, for m = 2, 3 for almost all (a, b) with ab ≠ 0, these equations have no nontrivial solutions. We also give infinitely many explicit examples when nontrivial solution does not exist.References
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