Timeline for Congruences between CM and non-CM modular forms
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jan 28, 2020 at 21:14 | comment | added | Filippo Alberto Edoardo | @DavidLoeffler Although somehow in an orthogonal direction, I wanted to point out a recent common paper with Nicolas Billerey jtnb.centre-mersenne.org/article/JTNB_2018__30_2_651_0.pdf for which your question has somehow been inspirational. We start with a residually dihedral form and, at the cost of putting dome $p$ in the, level we produce a CM form (in char. 0) which is congruent to the first, optimising level and weight. In your setting all forms have prime-to-$p$ level, so it is not the same problem. | |
| Jun 3, 2013 at 1:11 | vote | accept | David Loeffler | ||
| Jun 2, 2013 at 23:38 | answer | added | Orac | timeline score: 8 | |
| Jun 2, 2013 at 23:34 | comment | added | Noam D. Elkies | Um, never mind: this $\ell$ is not $1 \bmod p$ because I used the same prime for $\ell$ and $p$. Anyway, see what the tables turn up; there aren't that many CM forms to try for $g$. | |
| Jun 2, 2013 at 23:29 | comment | added | Noam D. Elkies | The answer to (2) seems to be Yes too. The first candidate for $N\ell$ is $98$, with $N = 49$ the conductor of the curve $[1,-1,0,-2,-1]$ with CM by ${\bf Z}[(1+\sqrt{-7})/2]$, and $\ell = 2$. There's an isogeny class of (non-CM) elliptic curves of conductor $98$, namely the ${\bf Q}(\sqrt{-7})$ twists of the curves of conductor $14$, such as $[1, 1, 0, -515, -4717]$. And indeed the modular forms are congruent $\bmod 2$, because each curve has a rational $2$-torsion point. (Or did you mean to require $p$ odd?) | |
| Jun 2, 2013 at 23:27 | comment | added | David Loeffler | Yes, (1) was a silly question, sorry! For (2), I did do some computer checks, and your question prompted me to make some more; at the second attempt I found an example with g the CM form of level 32, N = 17, and p = 5, but in fact for the special case I have in mind it would suffice to consider ell such that ell = 1 mod p, and I didn't find any examples where this is the case. | |
| Jun 2, 2013 at 23:24 | history | edited | David Loeffler | CC BY-SA 3.0 |
added requirement that ell is 1 mod p
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| Jun 2, 2013 at 22:27 | history | edited | David Loeffler | CC BY-SA 3.0 |
question (1) was silly, edited it out
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| Jun 2, 2013 at 22:06 | comment | added | Noam D. Elkies | (1) should be easy; for example, let $g$ be the modular form of level $N=32$ corresponding to the CM elliptic curve $y^2=x^3-x$, and take $p=3$ or $p=5$ so there are plenty of non-CM elliptic curves with the same $p$-torsion structure. For (2), did you trawl the known tables of modular forms? | |
| Jun 2, 2013 at 21:39 | history | asked | David Loeffler | CC BY-SA 3.0 |