TatesTheorem - crowlogic/arb4j GitHub Wiki

Tate's theorem, elliptic curves, modular forms, and Bessel function are related to the Birch and Swinnerton-Dyer (BSD) conjecture in various ways:

  1. Tate's Theorem: Tate's theorem about the points of finite order on the Néron model of an elliptic curve is a fundamental result in the study of elliptic curves. The Néron model is crucial for understanding the local and global arithmetic of elliptic curves. In the context of the BSD conjecture, understanding the structure of the group of rational points and the Tate-Shafarevich group is paramount, and Tate's work provides some of the foundational tools for this study.

  2. Modularity Theorem: As mentioned earlier, every elliptic curve over (\mathbb{Q}) is modular. This means the L-series of an elliptic curve corresponds to the L-series of a modular form. This connection allows us to harness the deep theory of modular forms when studying elliptic curves. The L-function's behavior, especially its order of vanishing at (s=1), plays a pivotal role in the BSD conjecture.

  3. Bessel Functions: As discussed, Bessel functions come into play in specific contexts, especially when relating the Fourier coefficients of modular forms or when studying certain integral representations associated with the L-values of elliptic curves. Such integral representations, involving Bessel functions, can be used to study the central values of L-functions, which is directly relevant to the BSD conjecture.

In essence, while the Birch and Swinnerton-Dyer conjecture is a statement about the relationship between the arithmetic of elliptic curves (the number of rational points) and their associated L-functions, the tools and techniques from the study of Tate's theorem, modular forms, and the theory surrounding Bessel functions provide a deep arsenal of methods to approach, study, and hopefully one day solve the conjecture.

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