The Birch–Swinnerton-Dyer conjecture is one of the most famous open problems in modern number theory; this is reflected by its inclusion in the Clay Mathematics Insti-tute million-dollar problems. The conjecture asserts that the rank of an abelian variety can be recovered from its L-function. In this thesis, we examine some of the conse-quences that are predicted by the Birch–Swinnerton-Dyer conjecture with the aid of Galois representations.
The first consequence is the parity conjecture: this states that the expected sign of the functional equation, known as the root number, should control the rank modulo 2; i.e. whether it is odd or even. We derive explicit formulae for the root number in terms of Jacobi symbols, as well as their generalisation to twisted root numbers. This is a very useful tool for numerically verifying the Birch–Swinnerton-Dyer conjecture and we give worked examples of computing the root number associated to the Jacobian of a hyperelliptic curve. As an application, we give sufficient criteria for an abelian variety such that every quadratic twist has infinitely many rational points, assuming the parity conjecture.
If one combines the Birch–Swinnerton-Dyer conjecture with a conjecture of Deligne–Gross, then one can obtain a generalised version concerning twisted L-functions. One can then use tools from representation theory to give predictions about: orders of van-ishing of the twisted L-functions; the corank of the ∞-Selmer group; and the existence of certain extensions where high orders of vanishing of the (untwisted) L-function al-ways occur, independently of the abelian variety.
Finally, we investigate the classical problem of distinguishing conjugacy classes of Frobenius elements in images of Galois representations. Using elliptic curves as the source of our Galois representations, we present two algorithms to distinguish between conjugacy classes of matrix groups in a small number of situations.
Original language | English |
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Award date | 2018 |
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