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Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
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CLASS NUMBERS OF IMAGINARY QUADRATIC FIELDS
, 2003
"... 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 GrossZagier, the task was a finite decision problem for any N. ..."
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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 GrossZagier, 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 GoldfeldOesterlé work, which used an elliptic curve Lfunction with an order 3 zero at the central critical point, to instead consider Dirichlet Lfunctions with lowheight 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 MontgomeryWeinberger. Our method is still quite computerintensive, 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.
Rabinowitsch Revisited
"... 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 wellknown (see [5]) that if the class number of some imaginary quadratic field with large discriminant ..."
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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 wellknown (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 Lfunction which is very close to 1). Thus Rabinowitsch's result can be informally stated as "n
Curves Dy 2 = x 3 − x of odd analytic rank
, 2002
"... 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 Lfunction of ED has sign −1, and thus odd analytic rank ran(ED), if and only if D is congruent to ..."
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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 Lfunction 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 SwinnertonDyer 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 nontorsion 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.
Elliptic curves, Lfunctions, and CMpoints
, 2002
"... this paper. The first four sections of the paper provide the standard background about elliptic curves and Lseries. 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 disc ..."
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this paper. The first four sections of the paper provide the standard background about elliptic curves and Lseries. 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 Lseries of elliptic curves have good analytic properties as predicted by the generalized TaniyamaShimura 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 Lseries: the rank of the MordellWeil group is equal to the order of vanishing of Lseries at the center (by the Birch and SwinnertonDyer 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
Arithmetic Properties of Class Numbers of Imaginary Quadratic Fields
, 2006
"... Under the assumption of the wellknown 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 cryptograp ..."
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Under the assumption of the wellknown 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
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"... A nearly zerofree region for L(s, χ), and Siegel’s theorem We used positivity of the logarithmic derivative of ζq to get a crude zerofree region for L(s, χ). Better zerofree regions can be obtained with some more effort by working with the L(s, χ) individually. The situation is most satisfactory ..."
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A nearly zerofree region for L(s, χ), and Siegel’s theorem We used positivity of the logarithmic derivative of ζq to get a crude zerofree region for L(s, χ). Better zerofree regions can be obtained with some more effort by working with the L(s, χ) individually. The situation is most satisfactory for