Even if they're significant gadgets within the conception of diophantine equations, the zeta-functions of Hasse-Weil will not be good understood. One huge classification of types whose zeta-functions are might be nearby are these hooked up to discrete teams, generically known as Shimura types. The innovations concerned are tricky: illustration concept and harmonic research; the hint formulation and endoscopy; intersection cohomology and $L2$-cohomology; and abelian types with advanced multiplication.The least difficult Shimura forms for which all attendant difficulties take place are these hooked up to unitary teams in 3 variables over imaginary quadratic fields, stated during this quantity as Picard modular surfaces. The participants have supplied a coherent and thorough account of valuable rules and strategies, a lot of that are novel and never formerly released.

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