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Let $f(z) = \sum_{n=1}^\infty a(n) e^{2i \pi nz}$ be an eigenform of $S_k(\Gamma_0(N))$. Since the Hecke operator acts by $T_p f = a_p f$ the Riemann hypothesis for $f$'s L-function is $$ \!\!\! \!\!\! \!\!\! \!\!\!(\sum_{n=1}^\infty a_n n^{-s})^{-1} =\prod_p (1-a(p) p^{-s}+p^{k-1-2s}) =\frac{\displaystyle[\prod_p (1-T_p p^{-s}+p^{k-1-2s})] f(z)}{f(z)} \tag{1}$$ converges and is analytic for $\Re(s) > k/2$.

Then, the Riemann hypothesis for all the eigenforms of $S_k(\Gamma_0(N))$ is that for any $f \in S_k(\Gamma_0(N))$, $$[\prod_p (1-T_p p^{-s}+p^{k-1-2s})] f(z) \quad \text{ is analytic for } \Re(s) > k/2\tag{2} $$

Questions :

This suggests to define a Riemann hypothesis for the Hecke operators themselves, and I would like to know if there is a well-known way to think about that.

  • The RH for the Hecke operators could be the convergence in operator norm of $\lim_{x \to \infty} \prod_{p \le x} (1-T_p p^{-s}+p^{k-1-2s}),$ for $\Re(s) > k/2$, with the norm coming from the Petersson inner product.

  • But there are other possible norms, for example $\langle f,g\rangle = \int_0^\infty f(ix) \overline{g(ix)} x^{2 \sigma -1}dx$. Indeed, the choice of the norm is a major problem : when defining $\displaystyle T_p f(z) = p^{k-1}\sum_{ad =p, b \bmod p} d^{-k} f(\frac{az+b}{d}) \tag{3}$ the statement in $(1)$ works the same way whenever $f(z) = \sum_{n=1}^\infty a_n e^{2i \pi n z}$ and $\sum_{n=1}^\infty a_n n^{-s} = \prod_p (1+a_p p^{-s}+ p^{k-1-2s})^{-1}$, no matter that $f$ is modular or not, so that in general $f$ doesn't have a Riemann hypothesis (nor the $T_p$ operator acting on it). So we really need a norm and a statement specific to modular forms, for example about $\prod_p (1-T_pp^{-s}+\langle p \rangle p^{k-1-2s})$ (which works for $S_k(\Gamma_1(N))$ too).

    For short, RH is believed true for Dirichlet series with Euler product and functional equation. $\prod_p (1-T_p p^{-s}+p^{k-1-2s})$ is just an Euler product. So hopefully, we only need to add a reference to the modularity (implying the functional equation) to make a viable RH statement.

  • Assuming we solved that part, the Hecke operators depend on $N$ only for finitely many $p$, so can we expect the statement to imply the Riemann hypothesis for $\displaystyle\bigcup_N S_k(\Gamma_0(N))$ ? In that case, what's about the dependence on $k$ ?

    Also looking at the weight-$\frac{1}{2}$ forms $\sum_{n \ge 1}^\infty \chi(n) e^{2i \pi n^2 z}$ could help.

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    $\begingroup$ I'm not sure I understand the question, but note that Hecke did not have the Petersson inner product to work with so was not able to diagonalize the space. All his work is in terms of operators. $\endgroup$
    – Stopple
    Commented Jul 6, 2017 at 20:08
  • $\begingroup$ @Stopple : For computing the operator norm coming from the Petersson inner product we need a basis of $S_k(\Gamma_0(N))$, so it isn't really different from $(1)$ and $(2)$. But changing the norm leads to a problem : the statement works the same way for functions having no RH. There is also the possibility to introduce things like $\langle p \rangle$ to make it specific to modular forms. Thus my question : what is the best way to think about all this, and is there a nice statement deserving to be called the RH for the Hecke operators ? $\endgroup$
    – reuns
    Commented Jul 6, 2017 at 20:26
  • $\begingroup$ Have you tried to consider the automorphism group of all the Hecke operators endowed with a suitable structure and to show that the analogue of RH you conjecture to exist is a property preserved under its action? $\endgroup$
    – Sylvain JULIEN
    Commented Jul 6, 2017 at 20:54
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    $\begingroup$ It is definitely not a good approximation to reality to suggest that Dirichlet series with Euler products and analytic continuation and functional equation satisfy a Riemann Hypothesis: Landau already had the example of $\zeta(2s)\zeta(2s-1)$ or similar. Also, to claim an RH for Dirichlet series taking values in Hecke operators would be asking for RH for all newforms and oldforms and linear combinations (with the same eigenvalues), which certainly could fail, without further discrimination. $\endgroup$
    – paul garrett
    Commented Jul 6, 2017 at 21:55
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    $\begingroup$ Well, modularity in a too-amorphous sense is insufficient: e.g., for a quadratic field extension of $\mathbb Q$ with absolute ideal class group having a sufficiently large $2$-part, we can make that number of unramified Hecke characters whose Dirichlet series have the same functional equation, have Euler products, and are attached to modular forms for $GL(2)$. But then it is trivial to make real-linear combinations of three or more having a zero at any given (e.g., off-line) location. But this linear combination does not have an Euler product, of course. $\endgroup$
    – paul garrett
    Commented Aug 7, 2017 at 13:28

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