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fixed $\ell$ vs $p$

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edited Sep 25, 2021 at 7:56

"Assuming they're in an algebraic extension of $\mathbb{Q}$ can they grow exponentially with 𝑝?"?

I'm not sure this statement will have a truth value, because I suspect the assumption never occurs: non-geometric $\ell$-adic representations are fundamentally p$\ell$-adic analytic objects and I don't know of any mechanism which would force their Frobenius traces to be in $\overline{\mathbb{Q}}$.

[Here I'm assuming that your representations are irreducible; one can easily construct examples of extensions $0 \to \chi_1 \to V \to \chi_2 \to 0$ with $\chi_i$ geometric which are unram almost everywhere but non-geometric at $p$$\ell$. But then the traces are the same as $\chi_1 \oplus \chi_2$ so the question is not interesting.]

There are examples of irreducible 2-dim'l non-geometric representations arising from non-classical overconvergent modular forms. I tabulated a bunch of these in my first ever paper (paywall, arxiv version here). The result is a list of p$\ell$-adic numbers (computed modulo some high power of p$\ell$), which are the q-expansion coefficients, or equivalently Frob traces at the first few primes; and they don't like they come from $\mathbb{Q}$ in any recognisable way.

"Assuming they're in an algebraic extension of $\mathbb{Q}$ can they grow exponentially with 𝑝?"?

I'm not sure this statement will have a truth value, because I suspect the assumption never occurs: non-geometric representations are fundamentally p-adic analytic objects and I don't know of any mechanism which would force their Frobenius traces to be in $\overline{\mathbb{Q}}$.

[Here I'm assuming that your representations are irreducible; one can easily construct examples of extensions $0 \to \chi_1 \to V \to \chi_2 \to 0$ with $\chi_i$ geometric which are unram almost everywhere but non-geometric at $p$. But then the traces are the same as $\chi_1 \oplus \chi_2$ so the question is not interesting.]

There are examples of irreducible 2-dim'l non-geometric representations arising from non-classical overconvergent modular forms. I tabulated a bunch of these in my first ever paper (paywall, arxiv version here). The result is a list of p-adic numbers (computed modulo some high power of p), which are the q-expansion coefficients, or equivalently Frob traces at the first few primes; and they don't like they come from $\mathbb{Q}$ in any recognisable way.

"Assuming they're in an algebraic extension of $\mathbb{Q}$ can they grow exponentially with 𝑝?"?

I'm not sure this statement will have a truth value, because I suspect the assumption never occurs: non-geometric $\ell$-adic representations are fundamentally $\ell$-adic analytic objects and I don't know of any mechanism which would force their Frobenius traces to be in $\overline{\mathbb{Q}}$.

[Here I'm assuming that your representations are irreducible; one can easily construct examples of extensions $0 \to \chi_1 \to V \to \chi_2 \to 0$ with $\chi_i$ geometric which are unram almost everywhere but non-geometric at $\ell$. But then the traces are the same as $\chi_1 \oplus \chi_2$ so the question is not interesting.]

There are examples of irreducible 2-dim'l non-geometric representations arising from non-classical overconvergent modular forms. I tabulated a bunch of these in my first ever paper (paywall, arxiv version here). The result is a list of $\ell$-adic numbers (computed modulo some high power of $\ell$), which are the q-expansion coefficients, or equivalently Frob traces at the first few primes; and they don't like they come from $\mathbb{Q}$ in any recognisable way.

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created Sep 25, 2021 at 7:25

David Loeffler

"Assuming they're in an algebraic extension of $\mathbb{Q}$ can they grow exponentially with 𝑝?"?

[Here I'm assuming that your representations are irreducible; one can easily construct examples of extensions $0 \to \chi_1 \to V \to \chi_2 \to 0$ with $\chi_i$ geometric which are unram almost everywhere but non-geometric at $p$. But then the traces are the same as $\chi_1 \oplus \chi_2$ so the question is not interesting.]

There are examples of irreducible 2-dim'l non-geometric representations arising from non-classical overconvergent modular forms. I tabulated a bunch of these in my first ever paper (paywall, arxiv version here). The result is a list of p-adic numbers (computed modulo some high power of p), which are the q-expansion coefficients, or equivalently Frob traces at the first few primes; and they don't like they come from $\mathbb{Q}$ in any recognisable way.