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by number-sequence
3291 days ago
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Yes, elliptic curves are huge in cryptography, but they are also mathematically significant. They are the degree three nonsingular algebraic plane curves with at least one rational point. This makes them essentially "one step up" from the conic sections. The conic sections, of course, are well understood including the parametrization of their rational points. However, elliptic curves are much more subtle! We do not even have proven algorithms for determining the size of any elliptic curve's set of rational points, and in specific the algebraic rank of that set. The Birch and Swinnerton-Dyer conjecture is one of the millennium prize problems, and it relates elliptic curves' algebraic ranks to their analytic ranks. Elliptic curves are also very related to modular forms, and this connection is part of the theory that allowed Andrew Wiles to prove Fermat's Last Theorem. In the study of the rational solutions to integer polynomial equations, i.e. Diophantine Analysis, elliptic curves are one of the next stepping stones that must be more thoroughly understood before we can have a more complete understanding of polynomials in general. Their applications in cryptography and integer factorization are huge in applied mathematics, but they are also incredibly important subjects to fields like algebraic geometry, diophantine analysis, and of course, number theory. |
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