Galois theory of Mordell-Weil groups - Vavasour, Thomas (2019)
Let K/k be a finite Galois extension of number fields with Galois group G, and let E be an elliptic curve defined over k. In this thesis we study the problem of trying to determine the Zp[G]-module structure of the p-adic completion E(K)* = E(K) ⊗Z Zp from more easily calculated invariants of K/k and E/k. In the case where G has cyclic p-Sylow subgroup, a theorem of Yakovlev tells us that the cohomology of E(K)* determines a part of E(K)*, and in the Chapter 3 we study these groups by way of a control theorem describing the cokernel of the natural restriction maps on the p-primary Selmer groups. In Chapter 4 we develop the necessary representation theory of Zp[G]-lattices, that is Zp[G]-modules that are Zp-free, for certain specific groups G whose order is divisible by p precisely once. In particular we calculate their regulator constants which, by a theorem of Torzewski, gives us the necessary ingredient to fully determing E(K)*. In Chapter 5 we combine the results from the previous two chapters to prove various results allowing us to determine the Zp[G]-structure of E(K)* in specific cases. Finally, in Chapter 6 we illustrate these results with concrete examples.