Timeline for Riemann hypothesis for the Hecke operators and modular forms
Current License: CC BY-SA 3.0
22 events
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| Aug 30, 2017 at 0:35 | comment | added | reuns | @paulgarrett Hi. Don't you think there is a lot to say about that ? ${}{}{}{}$ | |
| Aug 7, 2017 at 15:09 | comment | added | reuns | @paulgarrett Say $F(s)= \prod_p \frac{1}{1-a(p) p^{-s}}$ has a completely multiplicative Euler product, then the RH for $F(s)$ is $\lim_{N \to \infty} \sum_{k=1}^N \mu(k) a(k)k^{-s} F(s) = 1$ for $\Re(s) > 1/2$. My trick is to replace $a(k)k^{-s} F(s)$ by $F_k(s) = \sum_{n=1}^\infty a(nk)(nk)^{-s}$. Then look at $\lim_{N \to \infty}\sum_{k=1}^N \mu(k) F_k(s)= a(1) \quad \text{for } \Re(s) > 1/2 \quad(eq.1)$. This statement works when $F$ is replaced by a linear combination of $L$-functions. With $F(s)=\zeta(s)+L(s,\chi)$ then $(eq.1)$ implies $\frac1{\zeta(s)}+\frac{1}{L(s,\chi)}$ is analytic | |
| Aug 7, 2017 at 13:28 | comment | added | paul garrett | 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. | |
| Aug 7, 2017 at 6:07 | comment | added | reuns | The idea is that for a general Fourier series $f(z)$ then $\lim_{N \to \infty} \int_0^1 |\sum_{k=1}^N \mu(k) f_k(ix)|^2 x^{\sigma-1}dx < \infty$ doesn't have any reason to converge, and to find how the modularity changes the deal and would make it convergent. Intuitively, it could lead to statements about $\mu(k)$ implying the GRH. Also, trying to prove the equivalent of the PNT in this setting should be interesting. | |
| Aug 7, 2017 at 6:00 | comment | added | reuns | @paulgarrett Hi, I left the Hecke operators alone, but I'm now interested in "RH for the Dirichlet series with functional equation". Say $F(s)= \sum_{n=1}^\infty a_n n^{-s},\Lambda(s)=\Gamma(s/2) q^{-s}F(s) = \varepsilon \Lambda(1-s)$. Then the RH is $\lim_{N \to \infty} \sum_{k=1}^N \mu(k) \sum_{n=1}^\infty a_{nk} (nk)^{-s} = a_1$ for $\Re(s) > 1/2$. On the modular side it becomes $\lim_{N \to \infty} \int_0^1 |\sum_{k=1}^N \mu(k) f_k(ix)| x^{\sigma-1}dx < \infty$ for $\sigma > 1/2$ where $f_k(z) = \sum_{n \ge 1} a_{nk} e^{2i \pi nk z}= \frac{1}{k}\sum_{b=0}^{k-1}f(z+\frac{b}{k})$ | |
| Jul 7, 2017 at 2:11 | history | undeleted | reuns | ||
| Jul 7, 2017 at 1:50 | history | deleted | reuns | via Vote | |
| Jul 6, 2017 at 22:45 | comment | added | paul garrett | @reuns, not to introduce a pole or singularity, but just meaningless off-line zeros, finite linear combinations of the sort you just mentioned can create such effects. Thus, I think, a good statement/conjecture should manage to avoid such things. E.g., only refer to newforms, which in various possible ways grabs the easy classical information at good primes, and dodges ambiguities at bad primes. But/and the latter must be dealt with somehow, and it cannot be trivial, I think. (And, again, for other classical groups, the multiplicity-one feature tends to be lost...) | |
| Jul 6, 2017 at 22:41 | comment | added | paul garrett | @reuns, seeing your last comment: at good primes, old forms and new forms give the same L-function, by anyone's definition. But/and if one defines an L-function simply as a Mellin transform, the bad-prime factors may vary (in essentially artifactual, as opposed to meaningful, ways). Then by deliberately arranging linear combinations to do mildly perverse things, RH can be (reasonably, unsurprisingly) violated. So, suitably interpreted, for $GL_n$ something like this could be plausible, by multiplicity-one. But the latter is known to fail (at least in naive forms) for other classical groups. | |
| Jul 6, 2017 at 22:37 | comment | added | paul garrett | @reuns, I'm not meaning to dispute the possible truth of RH for all cuspidal L-functions attached to GL(2), for example, ... but I am uneasy about an assertion for "Hecke operators", as (e.g.) a (commutative) algebra of (e.g., convolution) operators. If the scope of these operators were delimited somehow, I'd start believing in the possibility that an RH-type assertion was not immediately implausible. (Just being a bit paranoid here, maybe...) | |
| Jul 6, 2017 at 22:34 | comment | added | reuns | @paulgarrett "RH is true for all the eigenforms $f_j$ of $S_k(\Gamma_0(N))$" iff "for each $j$ : $[\prod_p (1-T_p p^{-s}+ p^{k-1-2s})] f_j(z)$ is analytic for $\Re(s) > k/2$" iff "for each $f \in S_k(\Gamma_0(N))$ : $[\prod_p (1-T_p p^{-s}+ p^{k-1-2s})] f(z)$ is analytic for $\Re(s) > k/2$". I don't follow why oldforms could be a problem (and I don't know the adelic or symplectic world). | |
| Jul 6, 2017 at 22:24 | comment | added | paul garrett | @reuns, I'm not sure that it works that way, but perhaps I just don't understand your intended point... That is, bad-prime factors can change things, especially "oldform" complications, by taking linear combinations of things that otherwise do have an RH (conjecturally). That is, multiplicities (which would appear for symplectic groups, for example) of Hecke eigenvalues create complications, to my perception. | |
| Jul 6, 2017 at 22:18 | comment | added | reuns | @paulgarrett I think you missed a point. Non eigenforms don't have a RH but it is not a problem because if $f \in S_k(\Gamma_0(N))$ then $f(z) = \sum_{j=1}^J c_j f_j(z)$ where $f_j(z)$ are eigenforms with L-function $F_j(s)$ so that $[\prod_p (1-T_p p^{-s}+ p^{k-1-2s})] f(z) = \sum_{j=1}^J c_j \frac{f_j(z)}{F_j(s)}$, which is the observation justifying the RH is a property of the Hecke operators, not only of the eigenforms. | |
| Jul 6, 2017 at 21:55 | comment | added | paul garrett | 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. | |
| Jul 6, 2017 at 21:05 | comment | added | reuns | @SylvainJULIEN you mean the modularity (or functional equation) of the functions onto which the Hecke operators act (I added a statement about this) | |
| Jul 6, 2017 at 21:04 | history | edited | reuns | CC BY-SA 3.0 |
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| Jul 6, 2017 at 20:54 | comment | added | Sylvain JULIEN | 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? | |
| Jul 6, 2017 at 20:26 | comment | added | reuns | @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 ? | |
| Jul 6, 2017 at 20:23 | history | edited | reuns | CC BY-SA 3.0 |
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| Jul 6, 2017 at 20:08 | comment | added | Stopple | 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. | |
| Jul 6, 2017 at 19:50 | history | edited | reuns | CC BY-SA 3.0 |
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| Jul 6, 2017 at 19:44 | history | asked | reuns | CC BY-SA 3.0 |