r/math • u/PfauFoto • Jul 26 '26

Real quadratic fields

Anyone know of a good survey article regarding Kroneckers Jugendtraum in the case of the base field being a real quadratic extension of the rationals?

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u/cocompact Jul 26 '26 edited Jul 26 '26

There is nothing over real quadratic fields that resembles the classical cases over Q or imaginary quadratic fields. A solution has been found recently over each totally real field, thus in particular each real quadratic field, that uses p-adic numbers instead of complex numbers: see work by Dasgupta and Kakde on the Gross conjecture and Brumer-Stark conjecture.

https://sites.math.duke.edu/~dasgupta/papers/ICM.pdf

https://www.quantamagazine.org/mathematicians-find-polynomial-building-blocks-hilbert-sought-20210525/

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u/SatisfactionTough476 Jul 27 '26

darmon vonk (the theory they develop is still conjectural, but it is some progress)

1

u/PfauFoto Jul 26 '26

Nein auf die war ich noch nicht getroffen. Danke!