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JoeMerchant [soylentnews.org] writes:

Elliptic Curves Yield Their Secrets in a New Number System

https://www.quantamagazine.org/elliptic-curves-yield-their-secrets-in-a-new-number-system-20230706/ [quantamagazine.org]

Caraiani and Newton achieved modularity — for all elliptic curves over about half of all imaginary quadratic fields — by figuring out how to adapt a process for proving modularity pioneered by Wiles and others to elliptic curves over imaginary quadratic fields.

The work is a technical achievement in its own right, and it opens the door to making progress on some of the most important questions in math in the imaginary setting.

In the late 1950s, Yutaka Taniyama and Goro Shimura proposed that there is a perfect 1-to-1 matching between certain modular forms and elliptic curves. The next decade Robert Langlands built on this idea in the construction of his expansive Langlands program, which has become one of the most far-reaching and consequential research programs in math.

If the 1-to-1 correspondence is true, it would give mathematicians a powerful set of tools for understanding the solutions to elliptic curves. For example, there’s a kind of numerical value associated with each modular form. One of math’s most important open problems (proving it comes with a million-dollar prize) — the Birch and Swinnerton-Dyer conjecture — proposes that if that value is zero, then the elliptic curve associated to that modular form has infinitely many rational solutions, and if it’s not zero, the elliptic curve has finitely many rational solutions.

But before anything like that can be tackled, mathematicians need to know that the correspondence holds: Hand me an elliptic curve, and I can hand you its matching modular form. Proving this is what many mathematicians, from Wiles to Caraiani and Newton, have been up to over the last few decades.

Their result provides a foundation for investigating some of the same basic questions about elliptic curves over imaginary quadratic fields that mathematicians pursue over the rationals and the reals. This includes the imaginary version of Fermat’s Last Theorem — though additional groundwork needs to be laid before that is approachable — and the imaginary version of the Birch and Swinnerton-Dyer conjecture.

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The elliptic curve equation used in Bitcoin's cryptography is called secp256k1 which uses this equation: y²=x³+7, a=0 b=7.

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