Degrees of strongly special subvarieties and the André-Oort conjecture.

We give a new proof of the André-Oort conjecture under the generalised Riemann hypothesis. In fact, we generalise the strategy pioneered by Edixhoven, and implemented by Klingler and Yafaev, to all special subvarieties. Thus, we remove ergodic theory from the proof of Klingler, Ullmo and Yafaev and replace it with tools from algebraic geometry. Our key ingredient is a lower bound for the degrees of strongly special subvarieties coming from Prasad’s volume formula for S-arithmetic quotients of semisimple groups.