Let $A$ be the set of modular forms $f=\sum a(n)q^n$ of weight $k$ on $\Gamma_0(N)$ with character $\chi$ whose coefficients $a(n)$ are in the ring of integers $\mathcal{O}$ of a fixed number field $F$. Let $\mathfrak{P}$ be an ideal of $F$, Serre obtained the following result : the set of primes $p\equiv 1\pmod{N\mathfrak{P}}$ such that $f|T_p\equiv 2f\pmod{\mathfrak{P}}$ for all $f$ in $A$ has positive density. My question is about this congruence, taking two co-prime integers $a$ and $b$, is it possible to find infinitely many primes $p\equiv a\pmod{b}$ for which $f|T_p\equiv 2f\pmod{\mathfrak{P}}$ ? (we may replace 2 by other elements of $\mathcal{O}/\mathfrak{P}$).

  • $\begingroup$ It means that the Fourier cefficients of the two modular forms are two-by-two congruent modulo the ideal considered. You can see the paper of J.P. Serre " congruences et formes modulaires". $\endgroup$ – Zakariae.B Feb 3 '17 at 16:19
  • $\begingroup$ If $f$ is a Hecke eigenform $T_p f = a(p) f$, it means that for any $\mathfrak{P}$ then $a(p) \equiv 2 \bmod \mathfrak{P}$ for infinitely many $p\equiv 1\pmod{N\mathfrak{P}}$ ? Why is it true ? $\endgroup$ – reuns Feb 3 '17 at 16:52
  • $\begingroup$ By using Chebotarev density theorem on the Galois representations associated to the modular form $f$ ! This is the beauty of modular forms !! $\endgroup$ – Zakariae.B Feb 4 '17 at 9:50

There are two questions, the one where you ask whether it is possible to find a $p$ in an arithmetic sequence s.t. $f|T_p \equiv 2 f$ and the one s.t. $f | T_p \equiv \alpha f$ for some $\alpha$ in the residue field.

Here I answer the first question, by the negative: no, it is not always possible.

Here is a counter-example: take $f=\Delta$, $\mathcal O=\mathbb Z$, $\mathfrak P = 3\mathbb Z$. Then as you say, by Serre's result, $f | T_p \equiv 2 f$ for a positive density, and actually for all primes $p$ such that $p \equiv 1 \pmod{3}$. But $f|T_p \equiv 0$ for all primes $p$ such that $p \equiv -1 \pmod{3}$. To see both assertion below, since $f$ is an eigenform, it suffices to check that $\tau(p) \equiv 1 + p \pmod{3}$, which is actually a well-known congruence, where $\tau(n)$ is the $n$-th coefficient of $f=\Delta$.

Now I claim that the answer to the second question is also negative.

Here is a counter-example: take $f=\Delta^3$, $\mathcal O=\mathbb Z$, $\mathfrak P = 2\mathbb Z$. Then for all odd prime $p$, one has $T_p f \equiv 0$ or $T_p f \equiv \Delta$, but never $T_p f \equiv \alpha f$ for $\alpha \in \mathbb F_2$. Thus any $(a,b)$ gives a counter-example.

To give a positive result, let me say that the answer to the second question is positive when $f$ is an eigenform for almost all Hecke operators (or even infinitely many, that's enough.)


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