We will study the connections in between algebraic number theory and one-dimensional schemes (curves). More specifically we will cover: integral closures, plane curves, factorisation of ideals, the discriminants, the ideal class group, projective curves, nonsingular complete curves, zeta-functions and the Riemann-Roch Theorem. If time permits we will also cover Frobenius morphisms and the Riemann hypothesis and further topics like the Jacobian variety and Galois representations.
Australian National University