Rank distribution in a family of cubic twists

In 1987, Zagier and Kramarz published a paper in which they presented evidence that a positive proportion of the even-signed cubic twists of the elliptic curve $x^3+y^3=1$ should have positive rank. We extend their data, showing that it is more likely that the proportion goes to zero

On Q-derived polynomials

It is known that Q-derived univariate polynomials (polynomials defined over Q, with the property that they and all their derivatives have all their roots in Q) can be completely classified subject to two conjectures: that no quartic with four distinct roots is Q-derived, and that no quintic with a triple root and two other distinct roots is Q-derived. We prove the second of these conjectures

On Mordell-Weil groups of elliptic curves induced by Diophantine triples

We study the possible structure of the groups of rational points on elliptic curves of the form y^2=(ax+1)(bx+1)(cx+1), where a,b,c are non-zero rationals such that the product of any two of them is one less than a square.Comment: 17 pages; to appear in Glasnik Matematicki 42 (2007

Lower order terms in the 1-level density for families of holomorphic cuspidal newforms

The Katz-Sarnak density conjecture states that, in the limit as the conductors tend to infinity, the behavior of normalized zeros near the central point of families of L-functions agree with the N -> oo scaling limits of eigenvalues near 1 of subgroups of U(N). Evidence for this has been found for many families by studying the n-level densities; for suitably restricted test functions the main terms agree with random matrix theory. In particular, all one-parameter families of elliptic curves with rank r over Q(T) and the same distribution of signs of functional equations have the same limiting behavior. We break this universality and find family dependent lower order correction terms in many cases; these lower order terms have applications ranging from excess rank to modeling the behavior of zeros near the central point, and depend on the arithmetic of the family. We derive an alternate form of the explicit formula for GL(2) L-functions which simplifies comparisons, replacing sums over powers of Satake parameters by sums of the moments of the Fourier coefficients lambda_f(p). Our formula highlights the differences that we expect to exist from families whose Fourier coefficients obey different laws (for example, we expect Sato-Tate to hold only for non-CM families of elliptic curves). Further, by the work of Rosen and Silverman we expect lower order biases to the Fourier coefficients in families of elliptic curves with rank over Q(T); these biases can be seen in our expansions. We analyze several families of elliptic curves and see different lower order corrections, depending on whether or not the family has complex multiplication, a forced torsion point, or non-zero rank over Q(T).Comment: 38 pages, version 2.2: fixed some typos, included some comments from Steven Finch which give more rapidly converging expressions for the constants gamma_{PNT}, gamma_{PNT,1,3} and gamma_{PNT,1,4}, updated reference

Finding rational points on bielliptic genus 2 curves

We discuss a technique for trying to find all rational points on curves of the form $Y^2 = f_3 X^6 + f_2 X^4 + f_1 X^2 + f_0$, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field Q(a). If each of these elliptic curves has rank less than the degree of Q(a) : Q, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over Q(a) on the elliptic curves, for which x is in Q. This in turn allows us to find all Q-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over Q), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over Q

Rational points on certain del Pezzo surfaces of degree one

Let $f(z)=z^5+az^3+bz^2+cz+d \in \Z[z]$ and let us consider a del Pezzo surface of degree one given by the equation $\cal{E}_{f}: x^2-y^3-f(z)=0$. In this note we prove that if the set of rational points on the curve $E_{a, b}:Y^2=X^3+135(2a-15)X-1350(5a+2b-26)$ is infinite, then the set of rational points on the surface $\cal{E}_{f}$ is dense in the Zariski topology.Comment: 8 pages. Published in Glasgow Mathematical Journa