Rachel Newton

COMPUTING THE CASSELS-TATE PAIRING ON THE 3-SELMER GROUP OF AN ELLIPTIC CURVE

We extend the method of Cassels for computing the Cassels-Tate pairing on the 2-Selmer group of an elliptic curve, to the case of 3-Selmer groups. This requires significant modifications to both the local and global parts of the calculation. Our method is practical in sufficiently small examples, and can be used to improve the upper bound for the rank of an elliptic curve obtained by 3-descent.

THE PROPORTION OF FAILURES OF THE HASSE NORM PRINCIPLE

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.

Transcendental Brauer groups of products of CM elliptic curves

Let L be a number field and let E/L be an elliptic curve with complex multiplication by the ring of integers O_K of an imaginary quadratic field K. We use class field theory and results of Skorobogatov and Zarhin to compute the transcendental part of the Brauer group of the abelian surface ExE. The results for the odd order torsion also apply to the Brauer group of the K3 surface Kum(ExE). We describe explicitly the elliptic curves E/Q with complex multiplication by O_K such that the Brauer group of ExE contains a transcendental element of odd order. We show that such an element gives rise to a Brauer-Manin obstruction to weak approximation on Kum(ExE), while there is no obstruction coming from the algebraic part of the Brauer group.

Strangely dual orbifold equivalence I

n this brief note we prove orbifold equivalence between two potentials described by strangely dual exceptional unimodular singularities of type E_{14} and Q_{10} in two different ways. The matrix factorizations proving the orbifold equivalence give rise to equations whose solutions are permuted by Galois groups which differ for different expressions of the same singularity.

Bad Reduction of Genus Three Curves with Complex Multiplication

Let C be a smooth, absolutely irreducible genus 3 curve over a number field M. Suppose that the Jacobian of C has complex multiplication by a sextic CM-field K. Suppose further that K contains no imaginary quadratic subfield. We give a bound on the primes \(\mathfrak{p}\) of M such that the stable reduction of C at \(\mathfrak{p}\) contains three irreducible components of genus 1.

Explicit local reciprocity for tame extensions

We consider a tamely ramified abelian extension of local fields. Tameness guarantees the presence in K of roots of unity of degree equal to the ramification index; we do not assume that K contains any extra roots of unity. Under these conditions, we give a method for the explicit computation of local reciprocity.

Non-ordinary curves with a Prym variety of low

If

p

\pi: Y \to X

-rank

is an unramified double cover of a smooth curve of genus

Shadow Lines in the Arithmetic of Elliptic Curves

g