Abstract not found

The classical class number problem of Gauss asks for a classification of all imaginary quadratic fields with a given class number N. The first complete results were for N = 1 by Heegner, Baker, and Stark. After the work of Goldfeld and Gross-Zagier, the task was a finite decision problem for any N. Indeed, after Oesterlé handled N = 3, in 1985 Serre wrote, “No doubt the same method will work for other small class numbers, up to 100, say.” However, more than ten years later, after doing N =5, 6, 7, Wagner remarked that the N = 8 case seemed impregnable. We complete the classification for all N ≤ 100, an improvement of four powers of 2 (arguably the most difficult case) over the previous best results. The main theoretical technique is a modification of the Goldfeld-Oesterlé work, which used an elliptic curve L-function with an order 3 zero at the central critical point, to instead consider Dirichlet L-functions with low-height zeros near the real line (though the former is still required in our proof). This is numerically much superior to the previous method, which relied on work of Montgomery-Weinberger. Our method is still quite computer-intensive, but we are able to keep the time needed for the computation down to about seven months. In all cases, we find that there is no abnormally large “exceptional modulus” of small class number, which agrees with the prediction of the Generalised Riemann Hypothesis.

Abstract not found

this paper we are primarily interested in further developing the theory of quadratic polynomials for which many of the small values are prime (rather than "all" as in Rabinowitsch's result). It is well-known (see [5]) that if the class number of some imaginary quadratic field with large discriminant is one then we will have an egregious counterexample to the Generalized Riemann Hypothesis (that is, a zero of the associated Dirichlet L-function which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n

this paper. The first four sections of the paper provide the standard background about elliptic curves and L-series. We will start with elliptic curves defined by Weierstrass equations, and address two arithmetic questions: to compute the Mordell Weil group (Lang's conjecture) and to bound the discriminant in terms of the conductor (Szpiro's conjecture). Then we assume that the L-series of elliptic curves have good analytic properties as predicted by the generalized Taniyama-Shimura conjecture. In other words, we always work on modular elliptic curves (or more generally, abelian varieties of GL 2 -type). Of course, over Q, such a conjecture has been proved recently by Wiles and completed by Taylor, Diamond, Conrad, and Brueil. Both arithmetic questions addressed earlier have their relation with L-series: the rank of the Mordell-Weil group is equal to the order of vanishing of L-series at the center (by the Birch and Swinnerton-Dyer conjecture); the discriminant is essentially the degree of the strong modular parameterization. The theory of complex multiplications then provides many examples of modular elliptic curves and abelian varieties of GL 2 -type, and the foundation for the theory of Shimura varieties

Abstract. For nonzero rational D, which may be taken to be a squarefree integer, let ED be the elliptic curve Dy 2 = x 3 − x over Q arising in the “congruent number ” problem. 1 It is known that the L-function of ED has sign −1, and thus odd analytic rank ran(ED), if and only if |D| is congruent to 5, 6, or 7 mod 8. For such D, we expect by the conjecture of Birch and Swinnerton-Dyer that the arithmetic rank of each of these curves ED is odd, and therefore positive. We prove that ED has positive rank for each D such that |D | is in one of the above congruence classes mod 8 and also satisfies |D | < 10 6. Our proof is computational: we use the modular parametrization of E1 or E2 to construct a rational point PD on each ED from CM points on modular curves, and compute PD to enough accuracy to usually distinguish it from any of the rational torsion points on ED. In the 1375 cases in which we cannot numerically distinguish PD from (ED)tors, we surmise that PD is in fact a torsion point but that ED has rank 3, and prove that the rank is positive by searching for and finding a non-torsion rational point. We also report on the conjectural extension to |D | < 10 7 of the list of curves ED with odd ran(ED)> 1, which raises several new questions.

Under the assumption of the well-known heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptography are also discussed. 1

A nearly zero-free region for L(s, χ), and Siegel’s theorem We used positivity of the logarithmic derivative of ζq to get a crude zero-free region for L(s, χ). Better zero-free regions can be obtained with some more effort by working with the L(s, χ) individually. The situation is most satisfactory for

Abstract not found

A nearly zero-free region for L(s, χ), and Siegel’s theorem We used positivity of the logarithmic derivative of ζq to get a crude zero-free region for L(s, χ). Better zero-free regions can be obtained with some more effort by working with the L(s, χ) individually. The situation is most satisfactory for