Edwards curves are curves of the form x^2 + y^2 = 1 + dx^2y^2. Edwards curves often find their use in cryptographic applications due to the computational advantages they provide but in this thesis...Show moreEdwards curves are curves of the form x^2 + y^2 = 1 + dx^2y^2. Edwards curves often find their use in cryptographic applications due to the computational advantages they provide but in this thesis we focus on their geometric and arithmetic properties.Show less
This thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more...Show moreThis thesis is about rational points on so-called del Pezzo surfaces, which are a certain type of surfaces with relatively simple geometry. The lower the degree of a del Pezzo surface, the more intricate is its geometry. Let S be a del Pezzo surface over a field k. It is known that if the degree of S is not 1 and S(k) is non-empty (with a mild extra condition for degree 2), the surface S is k-unirational, meaning that there is a dominant rational map from some projective space to S. If k is an infinite field, this implies that the set S(k) of k-rational points lies Zariski dense in S. But in general, we do not know whether unirationality holds when the degree is 1, and the answer to this question seems way out of reach. So if we want to prove the density of the k-rational points of del Pezzo surfaces of degree 1, we have to search for alternative methods. In this thesis, a result is proven that gives sufficient and necessary conditions for the Zariski density of the rational points on a certain family of del Pezzo surfaces of degree 1.Show less
We describe a method of bounding the Mordell–Weil rank of an elliptic curve E over a number field k. The result of this method may improve upon an upper bound from the p-Selmer group for some odd...Show moreWe describe a method of bounding the Mordell–Weil rank of an elliptic curve E over a number field k. The result of this method may improve upon an upper bound from the p-Selmer group for some odd prime number p and involves an expression for the Cassels–Tate pairing on X(E/k) in terms of certain local pairings, one for each place v of k, which we call Tate local pairings. For each odd prime number p we give explicit formulas for the Tate local pairings both in the case where all p-torsion of E is locally defined over the base field and for the more general case. We prove that in the case where all p-torsion is rational the formula for the general case also suffices. This means that the elements in the two formulas differ by the norm of some element. We conjecture which element this should be and prove our conjecture for small primes.Show less
