In this paper we will review the theory of local heights on curves and describe its relationship to the global height pairing on the Jacobian. The local results are all special cases of Néron’s theory , ; the global pairing was discovered independently by Néron and Tate , We will also discuss extensions of the local pairing to divisors of arbitrary degree and to divisors which are not relatively prime. The first extension is due to Arakelov ; the second is implicit in Tate’s work on elliptic curves . I have also included several sections of examples which illustrate the general theory.
KeywordsElliptic Curf Prime Divisor Degree Zero Local Pairing Local Height
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