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Published online by Cambridge University Press: 30 May 2025
This is an introduction to some aspects of the arithmetic of elliptic curves, intended for readers with little or no background in number theory and algebraic geometry. In keeping with the rest of this volume, the presentation has an algorithmic slant. We also touch lightly on curves of higher genus. Readers desiring a more systematic development should consult one of the references for further reading suggested at the end.
Let k be a field. For instance, k could be the field ℚ of rational numbers, the fieldℝ of real numbers, the field ℂ of complex numbers, the field ℚp of p-adic numbers (see [Koblitz 1984] for an introduction), or the finite field 𝔽q of q elements (see Chapter I of [Serre 1973]). Let K be an algebraic closure of K. A (geometrically integral, affine) plane curve X over K is defined by an equation f(x, y).
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