p-adic analytic constructions of algebraic points
Heilbronn Number Theory Seminar
5th December 2018, 4:00 pm – 5:00 pm
Howard House, 4th Floor Seminar Room
The theory of complex multiplication was developed by Kronecker but was already started by Eisenstein and even by Gauss. The aim of this program, known as Kronecker's Jugendtraum, is to generate all abelian extensions of a number field by adjoining "special values" of "periodic functions", which are known as singular moduli. In a similar vein, the construction due to Heegner of algebraic points on elliptic curves can be seen also as a 1-dimensional version of the previous construction (which would be 0-dimensional).
In the late 1990's Henri Darmon introduced a construction of p-adic analytic points on elliptic curves, known initially as Stark--Heegner points, which are conjectured to being defined over abelian extensions of real quadratic fields. Somewhat backwards from a historical perspective, Darmon--Vonk introduced in 2017 the corresponding analogue of singular moduli.
The aim of this talk is to present my own point of view regarding these constructions, which should be better suited to generalization.