Modularity and BSD: L-functions for elliptic curves

Kweku A. Opoku-Agyemang

Working Paper Class 21

This paper explores the connection between the arithmetic properties of an elliptic curve E over a number field K and the analytic behaviour of its Hasse–Weil L-function L(E, s). It shows that the rank of E(K), the group of rational points on E, equals the order of vanishing of L(E, s) at s = 1, and that the leading term of the Taylor expansion of L(E, s) at s = 1 is determined by finer arithmetic invariants of E over K. The proof uses a special case of the modularity theorem for elliptic curves, which was established by Andrew Wiles and others.

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Opoku-Agyemang, Kweku A. (2023). "On the non-existence of rational points on a family of elliptic curves arising from Fermat’s Last Theorem." Machine Learning X Doing Working Paper Class 16. Machine Learning X Doing.

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