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added requirement that ell is 1 mod p
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David Loeffler
David Loeffler
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Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible [EDIT: it definitely is possible], so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$$\ell$ be a prime not dividing $N$ that is 1 mod p and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?

Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible [EDIT: it definitely is possible], so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$ be a prime not dividing $N$ and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?

Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible [EDIT: it definitely is possible], so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell$ be a prime not dividing $N$ that is 1 mod p and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?

question (1) was silly, edited it out
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David Loeffler
David Loeffler
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Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible [EDIT: it definitely is possible], so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$ be a prime not dividing $N$ and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?

Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible, so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$ be a prime not dividing $N$ and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?

Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible [EDIT: it definitely is possible], so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$ be a prime not dividing $N$ and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?

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David Loeffler
David Loeffler
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Congruences between CM and non-CM modular forms

Let $g$ be a cuspidal modular eigenform of weight 2 and level $N$ that has CM, so comes from a Groessencharacter of an imaginary quadratic field $K$. Let $p$ be a prime not dividing $N$.

(1) Is it possible that $g$ can be congruent mod $p$ to an eigenform that does not have CM by $K$?

I suspect that this may be possible, so I'd also be interested in the following more specific question:

(2) Suppose $g$ is new of level $N$ and "$p$-isolated" (not congruent modulo $p$ to any other eigenform of that weight and level). Let $\ell \ne p$ be a prime not dividing $N$ and split in $K$, and let $g'$ be either of the two eigenforms at level $N\ell$ corresponding to $g$. Can there exist eigenforms of level $N\ell$, congruent to $g'$ mod $p$, that do not have CM by $K$?