engleski

High rank elliptic curves with prescribed torsion group

The group of an elliptic curve over the rationals is the product of the torsion group and r copies of infinite cyclic group. There are exactly 15 possible torsion groups, but little is known about which values of rank $r$ are possible. The conjecture is that rank can be arbitrary large, but it seems to be very hard to find examples with large rang. The current record is an example of elliptic curve over Q with rank >= 28, found by Elkies in May 2006. The problem of the construction of high-rank elliptic curves has some relevance for cryptography. Namely, the discrete logarithm problem for multiplicative group F_q^* of a finite field can be solved in subexponential time using the Index Calculus method. For this reason, it was proposed by Miller and Koblitz that for cryptographic purposes, one should replace F_q^* by the group of rational points E(F_q) on an elliptic curve. The main reasons why Index Calculus method cannot be applied on elliptic curves are that it is difficult to find elliptic curves with large rank, it is difficult to find elliptic curves generated by points of small height, and it is difficult to lift a point of E(F_p) to a point of E(Q). There is even a stronger conjecture that for any of 15 possible torsion groups T we have B(T)=\infty, where B(T)=sup {; rank(E(Q)) : torsion group of E over Q is T};. It follows from results of Montgomery and Atkin & Morain (motivated by finding curves suitable for the elliptic curve method of factorization) that B(T)>= 1 for all admissible torsion groups T. We improved this result by showing that B(T) >= 3 for all T. The information about current records for all admissible torsion groups can be found on the web page http://web.math.hr/~duje/tors/tors.html. In this talk, we will describe some recent improvements on lower bounds for B(T). The similar methods can be applied in construction of high-rank elliptic curves with some other additional properties. In particular, we will present results related to elliptic curves induced by Diophantine triples, i.e. curves of the form y^2=(ax+1)(bx+1)(cx+1), where a, b, c are non-zero rationals such that ab+1, ac+1 and bc+1 are perfect squares.

elliptic curves; rank; torsion group

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