lmfdb support message

[John Jones]

I thought I had approved the message below, but it looks like it didn't go through.  If it did, sorry for the duplicate.

===============================================================

Hi LMFDB Support,

I am enquiring about L-function 2-234446-1.1-c1-0-2 for even Analytic rank 4 Elliptic curve with LMFDB label 234446.a1 (Cremona label 234446a1). Here, the Graph of the Z-function along the critical line manifests "first wave" in Quadrant 1 to be positive; as "conjecturally" do all even Analytic rank 0, 2, 4, 6, 8... Elliptic curves. ****I realize that this is only a solitary example for Analytic rank 4 elliptic curve as "supporting evidence"....****

For odd Analytic rank 1, 3, 5, 7, 9...; all the Graphs of the Z-function along the critical line for Analytic rank 1 manifest "first wave" in Quadrant 1 to be positive. Then all the Graphs of the Z-function along the critical line for Analytic rank 3 manifest "first wave" in Quadrant 2 to be negative. Is there a (conjectural) pattern; viz, I think all the Graphs of the Z-function along the critical line for Analytic rank 3, 5, 7, 9, 11, 13... should manifest "first wave" in Quadrant 2 to be negative as well ALTHOUGH alternating Quadrant 1 +ve / Quadrant 2 -ve manifestation is also a possibility for various / different Analytic rank of 3, 5, 7, 9, 11, 13... elliptic curves.

Do you have any expert opinion on the above enquiry of mine and/or do you have an example(s) of Graphs of the Z-function along the critical line for Analytic rank 5 or higher????

Many thanks for letting me know.


Kind regards,
John Ting
University of Tasmania
Australia
Email: jyc...@utas.edu.au

[John Ting]

Professor John Jones (LMFDB Managing Editor)

School of Mathematical and Statistical Sciences

Arizona State University, USA

Hi Prof. Jones (and LMFDB Support),

Thank you for your email below to approve my message.

I apologize by not being more precise with respect to my original message enquiry on L-function 2-234446-1.1-c1-0-2 for even Analytic rank 4 Elliptic curve with LMFDB label 234446.a1 (Cremona label 234446a1). I am now letting you (as one of LMFDB Managing Editors) and LMFDB Support know about my imprecision.

I have INCORRECTLY used the term Quadrant 2, which should "traditionally" be Quadrant 4. This can be confusing as per [correctly] outlined in Page 61 of my [lengthy draft] 62-page research paper. I have hereby also attached this paper titled On the L-functions from Generalized Riemann hypothesis, Birch and Swinnerton-Dyer conjecture, and the Prime numbers from Polignac's and Twin prime conjectures for your perusal. See Remark A.1 reproduced below from Page 61 of my paper:

"...As observational LMFDB study, QI Z(t) positivity / QIV Z(t) negativity criterion in Graphs of Z-function for elliptic curves in 0 < t < infty range is conjecturally used to characterize odd Analytic rank [having Point Symmetry] elliptic curves and even Analytic rank [having Line Symmetry] elliptic curves. Those with Analytic rank 1 seem to manifest QI Z(t) positivity [viz, the location of "first wave", of varying 'height', is in positive Z(t) Quadrant I Graph of Z-function]. Those with Analytic rank 3 (and likely 5, 7, 9, 11, 13,...) seem to manifest QIV Z(t) negativity [viz, the location of "first wave", of varying 'height', is in negative Z(t) Quadrant IV Graph of Z-function]. *Another possibility is that elliptic curves with [different] odd Analytic rank may alternatingly manifest QI Z(t) positivity and QIV Z(t) negativity*. All even Analytic rank {0, 2, 4, 6, 8, 10...} elliptic curves [but with all Analytic rank 0 elliptic curves NOT having first nontrivial zero at t = 0 and of varying proximity to t = 0] seem to manifest QI Z(t) positivity with progressive 'flattening' of "first wave" at Origin point. An "exception" here is Analytic rank 0 [non-elliptic] Riemann zeta function / Dirichlet eta function in Figure 11 with (pseudo-inverted) "first wave" has instead manifest QIV Z(t) negativity. An observational example is Analytic rank 4 elliptic curve with LMFDB label 234446.a1 that manifest QI Z(t) positivity with impressive 'flattening' of "first wave" at Origin point. Additionally via various Incompletely Predictable "complex interactions", we intuitively expect frequency / complexity of nontrivial zeros (spectrum) and integer N values of conductor (or level) in L-functions of elliptic curves to be empirically correlated with Analytic rank 0, 1, 2, 3, 4, 5...".

Hope to get some expert feedback soon from anyone cleverer than me w.r.t. my message enquiry. Once again, my sincere apologies and many thanks for your help.

Kind regards,

John Ting

University of Tasmania

Email: jyc...@utas.edu.au / jyct...@gmail.com

[Edgar Costa]

The definition of Z-plot https://www.lmfdb.org/knowledge/show/lfunction.zfunction

makes clear that the choice of sign is arbitrary.

Explicitly, it says:

> The square root is chosen so that Z(t)>0 for sufficiently small t>0.

Hence, no conclusion can be deduced from Z(t)>0 for sufficiently small t>0, as it was chosen this way.

[John Ting]

Hi Costa and LMFDB Support,

From the definition of Z-plot https://www.lmfdb.org/knowledge/show/lfunction.zfunction: "....The square root is chosen so that Z(t) > 0 for sufficiently small t > 0. The multiset of zeros of Z(t) matches that of L(1/2+it) and Z(t) changes sign at the zeros of L(1/2+it) of odd multiplicity...". I agree with the "contextual" feedback from Costa that the choice of sign is arbitrary. But once the choice of sign has been arbitrarily chosen on (1) odd Analytic rank elliptic curves with this chosen sign being always NEGATIVE versus (2) even Analytic rank elliptic curves with this chosen sign being always POSITIVE; I still think in the (1) odd Analytic rank elliptic curves scenario there are differentiating "research" features as outlined next.

In particular, odd Analytic rank 1 elliptic curves manifest "first wave" in Quadrant I being positive but odd Analytic rank 3, 5, 7, 9... elliptic curves should manifest "first wave" in Quadrant IV being negative. This is only my intuitive opinion along the parallel / analogical thinking that "... even Analytic rank 0 elliptic curves have first nontrivial zero NOT located at t = 0 but all other even Analytic rank 2, 4, 6, 8, 10... elliptic curves have first nontrivial zero located at t = 0...".

In other words, I think odd Analytic rank 3, 5, 7... elliptic curves should NOT have "first wave" in Quadrant I being positive... although could some of them alternatingly "flip" to do so??? I think obtaining a Z-plot of nontrivial zeros from an Analytic rank 5 elliptic curve to check its "first wave" being in Quadrant IV [versus in Quadrant I] is a useful research exercise to at least create some conjectures that may "semi-qualitatively" characterize different odd versus even Analytic rank elliptic curves especially in the difficult Birch and Swinnerton-Dyer conjecture setting...

Kind regards,

John Ting

[David W. Farmer]

Dear John Ting,

I will give more details about Edgar's answer.

There is no meaning to whether Z(t) is positive or
negative when t is small. This is because there is an
arbitrary choice of sign (coming from a square root)
when the Z-function is constructed from the L-function.

There is no way to resolve that ambiguity. See
Principle 2.2 in this paper:
https://arxiv.org/abs/2211.11671

Your observations follow from the fact that the Z-function
of an elliptic curve is even (or odd), depending on whether
the rank of the elliptic curve is even (or odd), coupled
with the misunderstanding that there is meaning to the
choice of sign when the Z-function was defined.

Regards,

David Farmer

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[John Ting]

Hi David Farmer and Edgar Costa,

Firstly, very sorry to Edgar for carelessly [and rudely] using your surname "Costa" as first name in my previous email reply !

Thank you very much David for your helpful and insightful email reply below, and your brilliant arXiv research paper at https://arxiv.org/abs/2211.11671 (12 December 2022). .... I statistically """extrapolate""" and interpret Principle 2.2 in your paper as the number of nontrivial zeros in Z(t) plots from -infinity < t < 0 range VERSUS 0 < t < +infinity range will always be ".... PRECISELY equal to each other irrespective of even Analytic rank 0, 2, 4, 6, 8, 10... (manifesting Line symmetry) versus odd Analytic rank 1, 3, 5, 7, 9... (manifesting Point symmetry)".

I agree with David about the """strict""" interpretation that "....There is no meaning to whether Z(t) is positive or negative when t is small. This is because there is an arbitrary choice of sign (coming from a square root) when the Z-function is constructed from the L-function....".

I have carefully compose the following Remark A.1 reproduced below (in "lazy" Latex format) from Page 62 - 63 of my [draft] research paper. On further reflection, I (respectfully) think the "....arbitrary choice of sign (coming from a square root) when the Z-function is constructed from the L-function..." is simply not very applicable or relevant... <<<----- BEST read from the research paper itself... Hope to get more feedback soon... Kind regards, John Ting

\begin{rem}\label{rem-Q1Q2} \textbf{Conjecture on "Altered Q I Z(t) positivity / Q IV Z(t) negativity criterion in Graphs of Z-function" (observational LMFDB study):} Let Analytic rank of elliptic curves $E$ = $r$ consist of even $r$ = 0, 2, 4, 6, 8, 10... and odd $r$ = 1, 3, 5, 7, 9.... Note: $r$ = 0 for (non-elliptic) Riemann zeta function $\zeta(s)$ / Dirichlet eta function $\eta(s)$. Polar graphs e.g. all Analytically normalized $\displaystyle \sigma=\frac{1}{2}$-Critical Line Polar graphs of $E$, Polar graph Figure \ref{+-30Riemann} on $\displaystyle \sigma=\frac{1}{2}$-Critical Line for (non-elliptic) $\zeta(s)$ / $\eta(s)$, etc will manifest features of even functions [when having even $r$] and odd functions [when having odd $r$]. Caveat: The horizontal x-axis and vertical y-axis are arbitrarily chosen such that Line symmetry is [dependently] the horizontal x-axis for Polar graphs having even $r$, but Point symmetry is [independently] the Origin point for Polar graphs having odd $r$. In comparison, Line symmetry is [dependently] the vertical y-axis for Graphs of Z-functions having even $r$, but Point symmetry is also [independently] the Origin point for Graphs of Z-functions having odd $r$. Considering $0<t<+\infty$ range in plotted trajectory of Polar graph or Graph of Z-function, let distance $d$ = difference between $P_1$ (trajectory initially intersecting horizontal x-axis of Polar graph / vertical y-axis in Graph of Z-function) and $P_2$ (trajectory initially intersecting Origin point of Polar graph / Graph of Z-function). Then (i) $d = P_2 - P_1$ $\neq$ 0 for $r$ = 0 $\zeta(s)$ / $\eta(s)$ and $r$ = 0 $E$, and (ii) $d = P_2 - P_1$ = 0 for $r$ = 1, 2, 3, 4, 5... $E$ [with these findings being equally valid for $-\infty<t<0$ range].

      

We recall the parity of (simple) polynomial functions as being EITHER $\pm$ even functions OR $\pm$ odd functions: [I] e.g. $y = \pm x^{0, 2, 4, 6, 8, 10...}$ being even functions with corresponding entire functions of $-\infty<x<+\infty$ range being located in Quadrant I and II when "$y$ is a $+ve$ function" / in Quadrant III and IV when "$y$ is a $-ve$ function". [II] e.g. $y = \pm x^{1, 3, 5, 7, 9, 11...}$ being odd functions with corresponding entire functions of $-\infty<x<+\infty$ range being located in Quadrant I and III when "$y$ is a $+ve$ function" / in Quadrant II and IV when "$y$ is a $-ve$ function". Nomenclature: Let $y$ and its exponents be now denoted by $\pm$Z(t) and $r$, and we only analyze the $0<t<+\infty$ range on [so-called] "first wave" of plotted Z-function for $E$ [and $\zeta(s)$ / $\eta(s)$] which will represent in a \textit{de-facto} manner $+$Z(t) with even $r$ or odd $r$ [viz, Q I Z(t) positivity], and $-$Z(t) with even $r$ or odd $r$ [viz, Q IV Z(t) negativity].

Then the above nomenclature explicitly $\implies$ $\zeta(s)$ / $\eta(s)$ in Figure \ref{Riemann Spectrum} is an $-$Z(t) with even $r$ = 0 [viz, showing Q IV Z(t) negativity]. $E$ in Figure \ref{NTZ Spectrum 0} is an $+$Z(t) with even $r$ = 0 [viz, showing Q I Z(t) positivity]. $E$ in Figure \ref{NTZ Spectrum 1} is an $+$Z(t) with odd $r$ = 1 [viz, showing Q I Z(t) positivity]. $E$ in Figure \ref{NTZ Spectrum} is an $+$Z(t) with even $r$ = 2 [viz, showing Q I Z(t) positivity]. $E$ in Figure \ref{NTZ Spectrum 3} is an $-$Z(t) with odd $r$ = 3 [viz, showing Q IV Z(t) negativity]. $E$ in Figure \ref{NTZ Spectrum 4} is an $+$Z(t) with even $r$ = 4 [viz, showing Q I Z(t) positivity and prominent \textit{flattening} of "first wave" at Origin point]. We tentatively propose all $E$ with even $r$ should likely be represented by $+$Z(t) showing Q I Z(t) positivity. However, it is unclear without further research whether $E$ with higher odd $r$ $\geq 5$ should be of [same] $-$Z(t) showing Q IV Z(t) negativity OR should be of [alternating] $+$Z(t) showing Q I Z(t) positivity / $-$Z(t) showing Q IV Z(t) negativity.

Additionally via various Incompletely Predictable "complex interactions", we intuitively expect frequency / complexity of nontrivial zeros (spectrum) and integer $N$ values of conductor (or level) in L-functions of elliptic curves to be empirically correlated with Analytic rank 0, 1, 2, 3, 4, 5....\end{rem}

---

From: David W. Farmer <far...@aimath.org>
Sent: Wednesday, 7 August 2024 3:48 PM
To: John Ting <jyct...@gmail.com>
Cc: Edgar Costa <edg...@mit.edu>; John Jones <j...@asu.edu>; lmfdb-support <lmfdb-...@googlegroups.com>; John Ting <jyc...@utas.edu.au>
Subject: Re: lmfdb support message

 

> https://groups.google.com/d/msgid/lmfdb-support/CAJ7-7kwEqr-4MZF7twdQmr4V...@mail.gmail.com.
>
>

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[John Ting]

Dear John,

On August 9, 2024 I posted problem "Determining geometric nature of  Z(t) nontrivial zeros plots of even/odd Analytic rank elliptic curves" at Mathematics Stack Exchange. On August 13, 2024 Aphelli had cleverly noticed and reported an "inconsistency" to LMFDB feedback page in relation to L-function attached to the elliptic curve 5077.a1 (https://www.lmfdb.org/L/2/5077/1.1/c1/0/410) whereby you (John Jones) had opened GitHub ticket Inconsistent normalization of the Z-functions attached to elliptic curves of rank 3 #6131 .

The so-called "LMFDB normalization" of Z(t), which is defined as positive for small positive t, ""seems"" to be correct for elliptic curve 5077.a1 BUT this normalization ""seems"" to be inconsistently applied to L-functions attached to [I think, just about all other] elliptic curves of rank 3.

Could the following mathematical concept "Sign normalization" [DIFFERENT to "LMFDB normalization"] be applicable to all Analytic rank 0, 1, 2, 3, 4,5....? In other words, could Z(t) plot for elliptic curve 5077.a1 be somehow "incorrectly" plotted OR is a "rare" exception?? I will also share my idea at GitHub... Look forward to feedbacks from LMFDB Support. Kind regards, John Ting (Email: jyc...@utas.edu.au)

Denote r = Analytic rank. Then "LMFDB normalization" is different to "Sign normalization" for epsilon which we advocate to be represented by (1)^^{r-1} for even r with epsilon = 1 and (i)^^{r-1} for odd r with epsilon = i [that obeys root of unity (for i)]. Intuitively, one anticipate Sign changes to occur exactly when r = 1, 2 (mod 4) BUT: [I] For even r = 0, 2, 4, 6, 8...; 1^^{r-1} = (1)^^{-1}, (1)^^{1}, (1)^^{3}, (1)^^{5}, (1)^^{7}... = same +1 sign [of +1, +1, +1, +1, +1,...]. c.f. [II] For odd r = 1, 3, 5, 7, 9...; i^^{r-1} = (i)^^{0}, (i)^^{2}, (i)^^{4}, (i)^^{6}, (i)^^{8}... = alternating +/- 1 sign [of +1, -1, +1, -1, +1,...].

---

From: David W. Farmer <far...@aimath.org>

Sent: Friday, 9 August 2024 2:15 PM
To: John Ting <jyc...@utas.edu.au>
Cc: Edgar Costa <edg...@mit.edu>; John Jones <j...@asu.edu>; John Ting <jyct...@gmail.com>

Subject: Re: lmfdb support message

 

Since, as I said, there is nothing wrong with the data or graphs, and the information you seek about the choice of sign is not mathematically meaningful, we cannot help you any further.

You can examine the source code of the LMFDB if you wish:

https://github.com/LMFDB/lmfdb

Regards,

David

On Fri, 9 Aug 2024, John Ting wrote:

Dear David,

** My "non-research" statements or deductions below are respectful. I stand corrected and my apology in advance if I am wrong or misguided**.

In LMFDB website, I could not determine from the individual {elliptic curve page / its associated L-function page / Downloads} the specific information I am seeking regarding Z(t) plots of nontrivial zeros for various Analytic rank elliptic curves.

The sign (root number) of functional equation = epsilon = +1 [for even Analytic rank elliptic curves "with Line symmetry"] or -1 [for odd Analytic rank elliptic curves "with Point symmetry"].

Z(t) plots will "meaninglessly" oscillate infinitely many times above and below the horizontal axis t. But Z(t) plots do "geometrically" display in the {0 < t < infinity range} the "first sinusoidal wave" that could be determined by arbitrarily choosing epsilon^1/2 = +1 or -1 (for "CASE" even Analytic rank) and epsilon^1/2 = +i or -i (for "CASE" odd Analytic rank). I agree with you that "... in either "CASE", we make a choice, but that choice is not mathematically meaningful...".

From what you wrote in your email below, I came up with the following observational concepts. Analytic rank r of elliptic curves E consist of:
(1) even r = 0, 2, 4, 6, 8, 10... [with epsilon = 1, epsilon^1/2 = +1 or -1 that can arbitrarily be chosen or assigned to display Z(t) plots in two reciprocal manners "+1 Z(t)" or "-1 Z(t)"], and
(2) odd r = 1, 3, 5, 7, 9... [with epsilon = -1, epsilon^1/2 = +i or -i that can arbitrarily be chosen or assigned to display Z(t) plots in two reciprocal manners "+i Z(t)" or "-i Z(t)"].

All available LMFDB Z(t) graphs for even Analytic rank elliptic curves [of r = 0, 2, "solitary" 4] are displayed as if we made the "standardized" choice of epsilon^1/2 = +1; viz "+1 Z(t)". For if [some of] the choice we made had been  epsilon^1/2 = -1, then I think some of the Z(t) plots will be displayed in a reciprocal manner "upside down" reflected by horizontal axis t as "-1 Z(t)".

All available LMFDB Z(t) graphs for odd Analytic rank elliptic curves [of 1 and 3] are displayed as if choice of epsilon^1/2 = +i; viz "+i Z(t)" was made for Analytic rank 1 BUT epsilon^1/2 = -i; viz "-i Z(t)" was made for Analytic rank 3 OR vice versa OR the choice of epsilon^1/2 was "standardized" BUT Analytic rank 1, then 3, then 5, then 7... will [intrinsically] display "first sinusoidal wave" in an alternatingly manner.

I think my big question to LMFDB Support would be whether Z(t) plots for Analytic 1 versus Analytic 3 elliptic curve had or had not (somehow) utilize different choice of epsilon^1/2 = +i OR epsilon^1/2 = -i. Then seeing the "geometrical character" of any future Z(t) plots for Analytic rank 5 elliptic would also be interesting.

Kind regards,
John

 ______________________________________________________________________________________________________________________________________

From: David W. Farmer <far...@aimath.org>

Sent: Thursday, 8 August 2024 11:12 PM
To: John Ting <jyc...@utas.edu.au>
Cc: Edgar Costa <edg...@mit.edu>; John Jones <j...@asu.edu>; John Ting <jyct...@gmail.com>

Subject: Re: lmfdb support message  

Dear John,

It is the sign of the functional equation where we take the square root.

If that sign is 1, then the square root is either 1 or -1.

If that sign is -1 then the square root is either i or -i.

In either case, we make a choice, but that choice is not mathematically meaningful.

This has nothing to do with the size of the square root of numbers like 9 or 0.002.

Regards,

David

[David W. Farmer]

There is no need for this discussion: the normalization described
in the documentation is not (currently) correctly implemented in the
code. That is just a bug in the code.

That error will be corrected.

John Ting: there is nothing interesting here: please drop this topic.

On Tue, 13 Aug 2024, John Ting wrote:

> Dear John,
>
> On August 9, 2024 I posted problem "Determining geometric nature of  Z(t) nontrivial zeros plots of even/odd Analytic rank elliptic
> curves" at Mathematics Stack Exchange. On August 13, 2024 Aphelli had cleverly noticed and reported an "inconsistency" to LMFDB
> feedback page in relation to L-function attached to the elliptic curve 5077.a1 (https://www.lmfdb.org/L/2/5077/1.1/c1/0/410) whereby
> you (John Jones) had opened GitHub ticket Inconsistent normalization of the Z-functions attached to elliptic curves of rank 3 #6131 .
>
> The so-called "LMFDB normalization" of Z(t), which is defined as positive for small positive t, ""seems"" to be correct for elliptic
> curve 5077.a1 BUT this normalization ""seems"" to be inconsistently applied to L-functions attached to [I think, just about all other]
> elliptic curves of rank 3.
>
> Could the following mathematical concept "Sign normalization" [DIFFERENT to "LMFDB normalization"] be applicable to all Analytic rank
> 0, 1, 2, 3, 4,5....? In other words, could Z(t) plot for elliptic curve 5077.a1 be somehow "incorrectly" plotted OR is a "rare"
> exception?? I will also share my idea at GitHub... Look forward to feedbacks from LMFDB Support. Kind regards, John Ting (Email:
> jyc...@utas.edu.au)
>
> Denote r = Analytic rank. Then "LMFDB normalization" is different to "Sign normalization" for epsilon which we advocate to be
> represented by (1)^{r-1} for even r with epsilon = 1 and (i)^{r-1} for odd r with epsilon = i [that obeys root of unity (for i)].
> Intuitively, one anticipate Sign changes to occur exactly when r = 1, 2 (mod 4) BUT: [I] For even r = 0, 2, 4, 6, 8...; 1^{r-1} =
> (1)^{-1}, (1)^{1}, (1)^{3}, (1)^{5}, (1)^{7}... = same +1 sign [of +1, +1, +1, +1, +1,...]. c.f. [II] For odd r = 1, 3, 5, 7, 9...;
> i^{r-1} = (i)^{0}, (i)^{2}, (i)^{4}, (i)^{6}, (i)^{8}... = alternating +/- 1 sign [of +1, -1, +1, -1, +1,...].
>

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