The rank of elliptic curves

On the complexity of computing the 2-Selmer group of an elliptic curve

“…This method is certainly fast for small values of D; however it is not hard to see that its complexity is at least O(VD); see [1,11]. In the present paper we shall show that the "old-fashioned" technique, which uses the arithmetic of number fields, combined with a method derived from a paper of Brumer and Kramer [2] will determine the 2-Selmer group in our stated time. Our complexity is therefore much better than the complexity of the algorithm of Birch and SwinnertonDyer.…”

“…For each prime p E S U {<»} we let U p be the image of E(Q P )/2E(Q P ) in G p under the mapping (2). In [2] Brumer and Kramer showed that the Selmer group is the maximal subgroup of K(R,2), whose image under the natural map a p is contained in U p for all primes p e S U {<»}. Ostensibly, to use this method, one must first calculate E(Q P )/2E(Q P ) for each prime p e S U {<»}.…”

“…We note that the number of ideals p t dividing (2D) is O(logZ)). The number of elements in a basis of Cl [2] is O(log(h K )) = O(log(£>)). Hence the number of ideals that we need to check to be principal is a polynomial function in log/).…”

Explicit upper bound for the (analytic) rank of J0(q)

“…the order of vanishing at the central critical point of their Hasse-Weil L-functions, see [MSD]) satisfies rank a J 0 (q) ≤ C dim J 0 (q) for all prime numbers q, C being some absolute constant. It is our purpose here to show how to compute an admissible value of C. To put the result in perspective, recall that it is conjectured [Br1] that rankJ 0 (q) = rank a J 0 (q) ∼ 1 2 dim J 0 (q) (based on the consideration of the sign of the functional equation of automorphic L-functions), the first equality being the famous conjecture of Birch and Swinnerton-Dyer in this particular case. Assuming the Riemann Hypothesis for automorphic L-functions, Iwaniec, Luo and Sarnak have recently proved that one could take C = 99 100 ; the best known previously was C = 23 22 [KM1], or C = 1 (using also the Riemann Hypothesis for Dirichlet L-functions).…”

The Modular Approach to Diophantine Equations

“…Let E be a semi-stable elliptic curve and ∆ be its minimal discriminant. Brumer and Kramer conjectured [7] that if |∆| is a perfect p-th power for some prime p, then p ≤ 7 and E has a point of order p. Serre gave a proof of this in [29] that was dependent at the time on what is known as Serre's conjectures. This dependency has now been removed thanks to the work of Ribet and Wiles.…”