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Swinnerton cycles are a concept in mathematics related to elliptic curves. Named after Peter Swinnerton-Dyer, these cycles provide valuable information about the structure and behavior of elliptic curves over finite fields. They are closely associated with the conjecture formulated by Swinnerton-Dyer, now known as the Birch and Swinnerton-Dyer conjecture, which proposes a connection between the number of rational points on an elliptic curve and certain arithmetic properties of its associated L-series. Although this conjecture remains unsolved, research on Swinnerton cycles has led to numerous advancements in number theory and algebraic geometry. Their study continues to have a significant impact on modern mathematics.

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