and 290-conjecture

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If I remember correctly (1998/99 part III), it is easy to check that the sum of real points is a real point again, so certainly there is a natural addition on the real curve. Then it must be a simple computation that it is actually given by that mysterious rule, the third point on the line. But there might be a conceptual reason even for this.

I guess this is one of the best examples for the difference between “blind computations” and “conceptual reasons”.

]]>Gross actually only considered definite forms in his talk, to avoid these sorts of complications with the indefinite case.

]]>I nominate René Thom’s thesis for introducing a problem that had no right to be solved, and then solving it in fine form. It’s probably tied with Serre’s though…

]]>>The 290 theorem says a positive definite integral quadratic form is universal if it takes the numbers from 1 to 290 as values. A more precise version states that, if an integer valued integral quadratic form represents all the numbers 1, 2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290 (sequence A030051 in OEIS), then it represents all positive integers, and for each of these 29 numbers, there is such a quadratic form representing all positive integers with the exception of this one number.

]]>Évariste Galois.

]]>But, clearly, Serre’s thesis is the best there is.

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