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Chapter 5Complex multiplication andelliptic curves“What does it mean that those trees and mountains out there are notmagic but real?” I’d yell, pointing outdoors.“What?” they’d say.“It means that those trees and mountains out there are not magic but real.”“Yeah?”Then I’d say, “What does it mean that those trees and mountains aren’treal at all, just magic?”“Oh come on.”“It means that those trees and mountains aren’t real at all, just magic.”“Well which is it, goddammit!”“What does it mean that you ask, well which is it goddammit?” I yelled.“Well what?”“It means that you ask well which is it goddammit.”The Dharma BumsJack Kerouac [149]Contents5.1 Further background on elliptic curves . . . . . . . . . . . . . . . . . 1325.1.1 Morphisms between algebraic curves . . . . . . . . . . . . . . . . . . . 1325.1.2 Divisors of algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . 1345.1.3 Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1365.1.4 Reduction of elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . 1385.2 Elliptic curves over the complex numbers . . . . . . . . . . . . . . . 1395.2.1 Complex tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405.2.2 Orders in imaginary quadratic fields . . . . . . . . . . . . . . . . . . . 1435.2.3 Binary quadratic forms . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.3 Elliptic curves with complex multiplication . . . . . . . . . . . . . . 1475.3.1 Complex multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . 1475.3.2 Hilbert class polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . 1485.3.3 The main theorem of complex multiplication . . . . . . . . . . . . . . 149