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Let $\overline{\rho}_{\Delta,\ell}$ be the mod-$\ell$ representation associated to Ramanujan's $\Delta$-function. It is well-known that (the semisimplification of) this representation is reducible if, say, $\ell=5$ or $\ell=691$. Is there a general name for primes like this? Serre calls them (in a more general context) "exceptional primes," but the word exceptional always strikes me as vague. "Primes of residual reducibility"?

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(From memory so might be wrong): I thought that primes like ell=23 were also deemed "exceptional" because the image is smaller than expected. But in this case the Galois representation is irreducible. So if I've remembered correctly, then "residual reducibility" is conveying a different notion to "exceptional". – Kevin Buzzard Nov 27 '09 at 21:02
@buzzard: yes, that is the "more general context" I had in mind. – David Hansen Nov 27 '09 at 21:35
Not a very informative hit though! You'd be better off working in weight 16 if you actually want to see some examples... – Kevin Buzzard Nov 28 '09 at 7:57
up vote 4 down vote accepted

"Eisenstein" ?

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AHHHH, thank you - I knew there was already a word for this! – David Hansen Nov 27 '09 at 22:19
Ok, but why? Relationship to Eisenstein series? – Ilya Nikokoshev Nov 27 '09 at 23:25
Sure: Eisenstein series give rise to char 0 reducible repns. The modular form Delta is congruent mod 691 to the Eisenstein series E_{12} so it's "Eisenstein mod 691". – Kevin Buzzard Nov 28 '09 at 8:02
Is it still called Eisenstein prime if it is congruent to any Eisenstein series? What I have in mind is, say an elliptic curve of conductor $N$ with a reducible mod 13 representation. Then $\rho_{13}^{ss} = \chi_1 \oplus \chi_2$, and the modular form attached to this curve, $f_E \in S_2(\Gamma_0(N))$, is congruent to a modular form in $\Gamma_1(N)$. However there are no Eisenstein series in $S_2(\Gamma_0(N))$ that $f_E$ is congruent to. – Soroosh Aug 18 '10 at 20:45

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