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Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Cited by 29 (2 self)
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
Examples of genus two CM curves defined over the rationals
 Math. Comp
, 1999
"... Abstract. We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the wellknown example y 2 = x 5 − 1 we find 19 nonisomorp ..."
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Cited by 20 (1 self)
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Abstract. We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the wellknown example y 2 = x 5 − 1 we find 19 nonisomorphic such curves. We believe that these are the only such curves. 1.
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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Cited by 8 (0 self)
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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.
Unramified quaternion extensions of quadratic number fields
 J. Théor. N. Bordeaux
, 1997
"... The first mathematician who studied quaternion extensions (H8extensions for short) was Dedekind [6]; he gave Q ( (2 + √ 2)(3 + √ 6) ) as an example. The ..."
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Cited by 8 (8 self)
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The first mathematician who studied quaternion extensions (H8extensions for short) was Dedekind [6]; he gave Q ( (2 + √ 2)(3 + √ 6) ) as an example. The
CMFIELDS WITH RELATIVE CLASS NUMBER ONE
, 2005
"... We will show that the normal CMfields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CMfields with relative class number one are of degrees ≤ 96, and the CMfields with class number one are of degrees ≤ 104. By many ..."
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Cited by 2 (0 self)
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We will show that the normal CMfields with relative class number one are of degrees ≤ 216. Moreover, if we assume the Generalized Riemann Hypothesis, then the normal CMfields with relative class number one are of degrees ≤ 96, and the CMfields with class number one are of degrees ≤ 104. By many authors all normal CMfields of degrees ≤ 96 with class number one are known except for the possible fields of degree 64 or 96. Consequently the class number one problem for normal CMfields is solved under the Generalized Riemann Hypothesis except for these two cases.
The class number one problem for some nonabelian normal CMfields of degree 48
 Math. Comp
"... Abstract. We prove that there is precisely one normal CMfield of degree 48 with class number one which has a normal CMsubfield of degree 16: the narrow Hilbert class field of Q ( √ 5, √ 101,θ)withθ 3 − θ 2 − 5θ − 1=0. 1. ..."
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Cited by 1 (1 self)
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Abstract. We prove that there is precisely one normal CMfield of degree 48 with class number one which has a normal CMsubfield of degree 16: the narrow Hilbert class field of Q ( √ 5, √ 101,θ)withθ 3 − θ 2 − 5θ − 1=0. 1.
Algebraic Numbers of Small Weil’s height in CMfields: on a Theorem of Schinzel ∗
, 2009
"... Let K be a CMfield. A. Schinzel proved ([Sch 1973]) that the Weil height of nonzero algebraic numbers in K is bounded from below by an absolute constant C outside the set of algebraic numbers such that α  = 1 (since ..."
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Let K be a CMfield. A. Schinzel proved ([Sch 1973]) that the Weil height of nonzero algebraic numbers in K is bounded from below by an absolute constant C outside the set of algebraic numbers such that α  = 1 (since
unknown title
, 2003
"... The class number one problem for some nonabelian normal CMfields of degree ..."
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The class number one problem for some nonabelian normal CMfields of degree