Even if they're primary gadgets within the conception of diophantine equations, the zeta-functions of Hasse-Weil aren't good understood. One huge type of types whose zeta-functions are maybe nearby are these hooked up to discrete teams, generically referred to as Shimura kinds. The concepts concerned are tough: illustration thought and harmonic research; the hint formulation and endoscopy; intersection cohomology and $L2$-cohomology; and abelian types with advanced multiplication.The least difficult Shimura kinds for which all attendant difficulties happen are these hooked up to unitary teams in 3 variables over imaginary quadratic fields, pointed out during this quantity as Picard modular surfaces. The members have supplied a coherent and thorough account of valuable rules and strategies, lots of that are novel and never formerly released.
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