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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 12

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ANALYTIC CONTINUATION

Road Map

To count points on elliptic curves efficiently and to formulate the Birch–Swinnerton-Dyer Conjecture, we will package certain counts in the form of L-functions, using ideas from chapter 11. Then we will have to study the properties of the L-functions. They are functions of a complex variable that are differentiable in a certain strong sense, earning them the epithet “analytic.” In this chapter, we will explain what analytic functions are and explore some of their properties.

Every polynomial is “analytic.” So we begin by discussing polynomials and how all of their values can be determined if you know ...

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