I often notice papers and results that assume that the level is squarefree in the setting of modular forms, but have a hard time figuring how where this impacts or simplifies the argument. Is there in particular to deal with easier Hecke relations, or are there deeper reasons for it? (may it be also be interpreted or understood in the light of elliptic curves or other analogues?)
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$\begingroup$ This question is far too broad; voting to close. $\endgroup$– David LoefflerCommented Oct 7 at 11:20
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6$\begingroup$ There are several different reasons squarefreeness of the level can simplify things, and different ones are relevant for different problems. Beyond listing them, not much can be said. A few off the top of my head: The trace formula simplifies in squarefree level. Atkin-Lehner operations act transitively on the cusps, so the q-expansion at every cusp looks the same. For squarefree level the local representation at p is determined from the coefficient of $q^p$ in the $q$-expansion for primes dividing $p$. For squarefree level mod $p$ exponential sums show up in certain problems, not $p$-adic. $\endgroup$– Will SawinCommented Oct 7 at 12:38
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2 Answers
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There are several ways in which studying modular forms with squarefree level is "simpler" for general level. Here I assume trivial nebentypus. For instance:
- You do not see CM forms.
- You do not see quadratic twists.
- The associated Galois/automorphic representations have simpler inertial/ramification types. E.g., you never see supercuspidal components in the automorphic representations.
- It's easier to isolate the newforms if one is using, say, trace formula methods.
- The only elliptic curves you get are semistable ones.
- The modular curves $X_0(N)$ are to some extent easier to understand.
- The connection with quaternion algebras (Jacquet-Langlands correspondence) is simpler/more complete.
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To my mind, the main impact of square-free level, at least with trivial character, is that (by the Borel-Casselman-Matsumoto theorem) the local representations are all sub-or-quotient repns of unramified principal series... so have very concrete models. In particular, none of the local repns can be supercuspidal, so no fancier modeling is necessary (although, by this year, we do have concrete models...)
Even with non-trivial character, at the all-but-finitely-many places where the character is unramified, the local repn is, again, a sub/quot of an unramified principal series.
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$\begingroup$ At least for me, it's also useful that for squarefree level (+ trivial character), the local level tells you the local representation type, so you don't have to do any work to separate out ramified p.s. vs special vs supercuspidal. $\endgroup$– KimballCommented Oct 8 at 15:14
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$\begingroup$ PS I didn't try to order the list in my answer by importance. $\endgroup$– KimballCommented Oct 8 at 15:15