Timeline for Existence of rational points on some genus 3 curves
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2023 at 20:46 | history | edited | Jeremy Rouse | CC BY-SA 4.0 |
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| Apr 14, 2023 at 10:42 | comment | added | Michael Stoll | The L-factor at $p = 11$ is also irreducible, and the number fields defined by the two polynomials are linearly disjoint. This implies that the endomorphism ring of the Jacobian is just the ring of integers, so there is also no extra structure there. | |
| Apr 14, 2023 at 10:36 | comment | added | Michael Stoll | The L-factor of the first curve at $p=5$ is irreducible, so its Jacobian is simple. In fact, since no (nontrivial) quotient of two of the roots of the L-factor is a root of unity, it is even absolutely simple. This means that no reduction to a lower-genus curve or lower-dimensional abelian variety is possible. | |
| Apr 13, 2023 at 17:33 | comment | added | Bogdan Grechuk | Thank you! I voted your answer up, did not accept because the second equation is open. It would be useful if you would (i) provide a reference to the result that if a curve has a non-trivial Cassels-Tate pairing, this shows that it is locally solvable everywhere, but has no rational points. And (ii) give a hint how you computed this Cassels-Tate pairing, is there a CAS code that can do it? | |
| Apr 13, 2023 at 12:53 | history | answered | Jeremy Rouse | CC BY-SA 4.0 |