Timeline for Congruence Number of 197A1
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 24, 2023 at 20:55 | comment | added | Jeremy Rouse | In the case that $r$ is a prime number, this is a consequence of the Deligne-Serre lifting lemma (see Theorem 1.2 and Corollary 1.3 of the document here). In the case that $r$ is composite, maybe what I said is a little bit too naive (there might be a different eigenform $h_{i}$ for each prime factor of $r$ with $f \equiv h_{i} \pmod{\mathfrak{r}_{i}}$). | |
| Jun 24, 2023 at 5:22 | comment | added | Adithya Chakravarthy | @JeremyRouse A clarification about your comment: why is it true that if there is a form $g$ with integer coefficients (not necessarily an eigenform) such that $f \equiv g$ mod $r$, then there must exist an eigenform, say $h$, with algebraic integer coefficients such that $f \equiv h$ mod $\mathfrak{r}$? (Here $\mathfrak{r}$ is an appropriate ideal in the ring of integers of the coeff field of $h$.) That is, if $f$ is congruent to a cuspform $g$ with integer coeffs, why must $f$ also be congruent to an eigenform $h$ with algebraic integer coeffs? | |
| Aug 6, 2015 at 2:26 | vote | accept | Jeff H | ||
| Aug 6, 2015 at 2:24 | comment | added | Jeff H | @ABCDveve If you ask LMFDB for elliptic curves of conductor 197, there is exactly one. | |
| Aug 6, 2015 at 0:01 | answer | added | Jeremy Rouse | timeline score: 5 | |
| Aug 5, 2015 at 23:49 | comment | added | ABCDveve | It seems the LMFDB is broken again. I clicked on your LMFDB link and got to 197a1 page, but going to Related objects "Modular form 197.2a" gave the excuse: " We are very sorry. The sought space could not be found in the database. " How are you able to use LMFDB to verify this? | |
| Aug 5, 2015 at 19:19 | comment | added | Jeff H | Or I can delete this question and pretend I never asked it ;) | |
| Aug 5, 2015 at 19:17 | comment | added | Jeff H | @JeremyRouse Oh, of course! What a silly mistake on my part. If you'd like to elevate your comment to an answer, I'd be happy to accept it. | |
| Aug 5, 2015 at 19:15 | comment | added | Jeremy Rouse | I'm skeptical of your stated definition of the congruence number. According to Zagier, the congruence number is not the largest positive integer for which there is another integer coefficient eigenform congruent to it. It is the largest positive integer $r$ for which there is a form $g$ with integer coefficients in the $\mathbb{C}$-span of the other eigenforms with $g \equiv f \pmod{r}$. (Essentially, there is no eigenform with integer coefficients congruent to $f$, but there is another eigenform with algebraic integer coefficients, and the same "mod 10" reduction.) | |
| Aug 5, 2015 at 19:06 | history | edited | Jeff H | CC BY-SA 3.0 |
left out the important qualification that g must be an eigenform
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| Aug 5, 2015 at 18:56 | history | asked | Jeff H | CC BY-SA 3.0 |