Timeline for Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?
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Jul 16, 2022 at 21:53 | comment | added | user471019 | It's equation (2.8.10) in Cremona. There is an additional factor of $2$ there. By the way, Magma tells me that the $L$-ratio is $1/4$. (It depends on how you define $\Omega_E$.) | |
Jul 16, 2022 at 19:04 | comment | added | Duality | I'm confused. For $p=2$, the elliptic curve has good reduction, so the denominator is smaller than $3$ ? Which prime $p$ you take ? | |
Jul 16, 2022 at 18:23 | comment | added | user471019 | In Cremona's book (somewhere around §2.8?), there is a proof using modular symbols that $L(E,1)/\Omega_E$ is a rational number with denominator bounded by $(p+1)-a_p$ for $p$ a prime of good reduction. Hence, approximating that number to this precision will give it you exactly. | |
Jul 16, 2022 at 17:22 | comment | added | Duality | If we admit $L(E,1)=0.65551438837・・・$ and $Ω(E)=5.24411510858・・・$ , from this, can we say $L(E,1)/Ω(E)=1/8$ without using computer ? | |
Jul 16, 2022 at 17:19 | history | edited | Michael Hardy | CC BY-SA 4.0 |
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Jul 16, 2022 at 13:34 | history | edited | Duality | CC BY-SA 4.0 |
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Jul 16, 2022 at 11:29 | comment | added | Chris Wuthrich | You can do so by using module symbols, as it is the value of the modular symbol (suitably normalised with respect to your period) evaluated at 0. See this question here for how this is done. Cremona's book is a good source for the basics of how to verify BSD for elliptic curves. | |
Jul 16, 2022 at 10:46 | history | edited | Duality | CC BY-SA 4.0 |
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Jul 16, 2022 at 9:12 | history | edited | Duality | CC BY-SA 4.0 |
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Jul 16, 2022 at 8:25 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
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Jul 16, 2022 at 8:03 | history | edited | Duality | CC BY-SA 4.0 |
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Jul 16, 2022 at 7:33 | history | asked | Duality | CC BY-SA 4.0 |