Even supposing they're critical items within the conception of diophantine equations, the zeta-functions of Hasse-Weil are usually not good understood. One huge type of types whose zeta-functions are probably close by are these hooked up to discrete teams, generically referred to as Shimura forms. The options concerned are tough: illustration thought and harmonic research; the hint formulation and endoscopy; intersection cohomology and $L2$-cohomology; and abelian types with complicated multiplication.The easiest Shimura types for which all attendant difficulties ensue are these connected to unitary teams in 3 variables over imaginary quadratic fields, said during this quantity as Picard modular surfaces. The members have supplied a coherent and thorough account of beneficial rules and strategies, a lot of that are novel and never formerly released.
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