Here we calculate the value distribution of the first derivative of characteristic polynomials of matrices from SO(2N + 1) at the point 1, the symmetry point on the unit circle of the eigenvalues of these matrices. The connection between the values of random matrix characteristic polynomials and values of the L-functions of families of elliptic curves implies that this calculation in random mat...

  In order to find an alternative source for the highly unsaturated fatty acids oil with lower cost, a marine trash fish Odonus niger has been analysed and characterised. The body weight and the corresponding liver weight of the fish were studied and the results showed that for one gram increase in the size of the animal, the liver weight can increase to 0.04 g. The oil yielding capacity of the...

We study the elliptic curve E given by y = x(x+1)(x+ t) over the rational function field k(t) and its extensions Kd = k(μd, t). When k is finite of characteristic p and d = p + 1, we write down explicit points onE and show by elementary arguments that they generate a subgroup Vd of rank d − 2 and of finite index in E(Kd). Using more sophisticated methods, we then show that the Birch and Swinner...

Building on our previous work on rigid analytic uniformizations, we introduce Darmon points on Jacobians of Shimura curves attached to quaternion algebras over Q and formulate conjectures about their rationality properties. Moreover, if K is a real quadratic field, E is an elliptic curve over Q without complex multiplication and χ is a ring class character such that LK(E,χ, 1) 6= 0 we prove a G...

Fix an elliptic curve E/Q, and assume the generalized Riemann hypothesis for the L-function L(E D , s) for every quadratic twist E D of E by D ∈ Z. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of E D. It follows from this that, for any unbounded increasing function f on R, the analytic rank an...

This article establishes new cases of the Birch and Swinnerton-Dyer conjecture in analytic rank 0, for elliptic curves over Q viewed over the fields cut out by certain self-dual Artin representations of dimension at most 4. When the associated L-function vanishes (to even order ≥ 2) at its central point, two canonical classes in the corresponding Selmer group are constructed and shown to be lin...

This paper is the second of a series devoted to the study of the rank of J0(q) (the Jacobian of the modular curve X0(q)), from the analytic point of view stemming from the Birch and Swinnerton-Dyer conjecture, which is tantamount to the study, on average, of the order of vanishing at the central critical point of the L-functions of primitive weight two forms f of level q (q prime). We prove tha...

A positive integer D is called a ‘congruent number’ if there exists a right triangle with rational sidelengths with area D. Over the centuries there have been many investigations attempting to classify the congruent numbers, but little was known until Tunnell [T] brilliantly applied a tour de force of methods and provided a conditional solution to this problem. It turns out that a square-free i...