Characterization of the torsion of the Jacobian of y²=x⁵+Ax and some applications

Abstract

1. Introduction. In [7] it is shown that for any quadruple of pairwise distinct elliptic curves E i , i = 1, 2, 3, 4, with j-invariant j = 0 there exists a polynomial D ∈ Z[u] such that the sextic twists of E i , i = 1, 2, 3, 4, by D(u) have positive ranks. A similar result was proved for quadruples of elliptic curves with j-invariant equal to 1728.These results have been generalized in [4] to curves of the form y 2 = x n + A, where n is divisible by an odd prime and A ∈ Z \ {0}. The main tool in the proof was a (partial) characterization of possible torsion subgroups of the Jacobian variety associated with the curve y 2 = x p + A, where p is an odd prime and A ∈ Z \ {0}.A natural question arises whether similar results could be obtained for ktuples (for appropriate k) of hyperelliptic curves of the form C : y 2 = x 5 +Ax (of genus 2), which are (in some sense) natural generalizations of elliptic curves with j-invariant equal to 1728. The quartic twist of the curve C by D has the form C D : y 2 = x 5 + AD 2 x. The octic twist of C by D is C D : y 2 = x 5 + ADx. In view of the results obtained in [7] and [4] we ask the following question.Question 1.1. Let C i : y 2 = x 5 +a i x where a i ∈ Z\{0} for i = 1, . . . , n, and suppose that C i = C j for i = j. What is the maximal number, say N , for which there exists a polynomial D ∈ Z [u] such that the Jacobian of the octic twist of C i by D has positive rank for all i = 1, . . . , N ? This is a difficult problem and we only give a lower bound on N . It is clear that to this end we need two things. We must be able to construct rational points on appropriate curves and show that these points lead to 2010 Mathematics Subject Classification: 11G05, 14G99.

On certain Diophantine systems with infinitely many parametric solutions and applications

“…In the cited paper, this result was used to show that if C i : y 2 = x n +a i , where a i ∈ Z\{0} are pairwise distinct, then there exists a polynomial D ∈ Z[t] such that the Q(t)-rank of the Jacobian variety Jac(C i,D ) is positive, where C i,D : y 2 = x n + a i D(t) for i = 1, 2, 3, 4. Similar results were proved in [8,9] and [3], where instead of f (x, y), we considered g(x, y) = (y 2 − x 3 )/x and g(x, y) = (y 2 − x 5 )/x respectively. In the light of these results, it is natural to ask what can be said about a general system of the form (1) h(x 1 , y 1 )…”

Variations on twists of tuples of hyperelliptic curves and related results

“…In the case of odd n it is possible to extend this result to four curves of the considered form. In [10] we also considered the octic twists of the hyperelliptic curves defined by the equation y 2 = x 5 +Ax and proved similar result. The questions leading to this results were motivated by the work of the second author concerned with the existence of simultaneous twists (of the same type) for tuples of elliptic curves [13,14].…”

“…Conjecture 6.4. Let a, b, c ∈ Z \ {0} and consider the hyperelliptic curves (10). Then the set of those d ∈ Q such that the Jacobian of the m-twist of the curve C i by d have positive rank for i = 1, 2, 3, is infinite.…”

“…Let a, b, c ∈ Z \ {0} and let m be an odd positive integer. Let us consider the hyperelliptic curves(10) C 1 :y 2 = x m 1 + a, C 2 : y 2 = x m + b, C 3 : y 2 = x m + c.Then there exist a rational function D m,m,2m ∈ Q(u, v, w) such that the Jacobians of the m-twists of the curves C 1 , C 2 and the Jacobian of the 2m-twist of the curve C 3 by D m,m,2m (u, v, w) have positive rank over the field Q(u, v, w).…”

Characterization of the torsion of the Jacobians of two families of hyperelliptic curves

“…We have In some sense the natural generalizations of elliptic curves with j -invariants 1728 and 0 are the hyperelliptic curves C n,A : y 2 = x n + Ax and C n,A : y 2 = x n + A (and their Jacobians J n,A and J n,A ) respectively. In [JU,Theorem 2.2] we proved that (1.3) J 5,A (Q) tors = J 5,A (Q) [2] for all A ∈ Q\{0}.…”