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3 Acta Arithmetica

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On congruent primes and class numbers of imaginary quadratic fields

100%

Bruin N. , Hemenway B.

Acta Arithmetica

2013

tom 159

nr 1

63-87

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate-Shafarevich groups of the associated elliptic curves. We also present a related criterion for primes p such that $16$ divides the class number of the imaginary quadratic field ℚ(√-p). Both results are based on descent methods. While we cannot show for either criterion individually that there are infinitely many primes that satisfy it nor that there are infinitely many that do not, we do exploit a slight difference between the two to conclude that at least one of the criteria is satisfied by infinitely many primes.

Rational points on twists of X₀(63)

80%

Bruin N. , Fernández J. , González J. , Lario J.-C.

nr 4

361-385

Descent via (3,3)-isogeny on Jacobians of genus 2 curves

80%

Bruin N. , Flynn E. , Testa D.

Acta Arithmetica

2014

tom 165

nr 3

201-223

We give a parametrization of curves C of genus 2 with a maximal isotropic (ℤ/3)² in J[3], where J is the Jacobian variety of C, and develop the theory required to perform descent via (3,3)-isogeny. We apply this to several examples, where it is shown that non-reducible Jacobians have non-trivial 3-part of the Tate-Shafarevich group.