**Professor (Grade 11) in Number Theory **

*Research Interests: * My research interests are in arithmetical algebraic geometry and computational number theory. In particular I work on elliptic curve descent calculations, and the construction of explicit elements in the Tate-Shafarevich group.

## Publications

Higher descents on an elliptic curve with a rational 2-torsion point

– Mathematics of Computation

(2017)

86,

2493

(doi: 10.1090/mcom/3163)

On genus one curves of degree 5 with square-free discriminant

– JOURNAL OF THE RAMANUJAN MATHEMATICAL SOCIETY

(2016)

31,

359

Visualizing elements of order $7$ in the Tate–Shafarevich group of an elliptic curve

– LMS Journal of Computation and Mathematics

(2016)

19,

100

(doi: 10.1112/S1461157016000243)

The proportion of plane cubic curves over Q that everywhere locally have a point

– International Journal of Number Theory

(2016)

12,

1077

(doi: 10.1142/s1793042116500664)

What is the Probability that a Random Integral Quadratic Form in n Variables has an Integral Zero?

– International Mathematics Research Notices

(2015)

2016,

3828

(doi: 10.1093/imrn/rnv251)

Ranks of quadratic twists of elliptic curves

– Publications Mathématiques de Besançon

(2015)

63

(doi: 10.5802/pmb.9)

On families of 9-congruent elliptic curves

– Acta Arithmetica

(2015)

171,

371

(doi: 10.4064/aa171-4-5)

Computing the Cassels-Tate pairing on the 3-Selmer group of an elliptic curve

– International Journal of Number Theory

(2014)

10,

1881

(doi: 10.1142/S1793042114500602)

On families of 7- and 11-congruent elliptic curves

– LMS Journal of Computation and Mathematics

(2014)

17,

536

(doi: 10.1112/S1461157014000059)

Minimal models for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}6$-coverings of elliptic curves

– LMS Journal of Computation and Mathematics

(2014)

17,

112

(doi: 10.1112/s1461157014000217)

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