New Rational Points of Algebraic Curves over Extension Fields
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Rational numberMany-sorted logicAbelsche MannigfaltigkeitPoint (geometry)MathematicsSet theoryField extensionFinitismusGroup actionGroup representationTensorAlgebraic number fieldKörper <Algebra>Structural stabilityGeometryVariety (linguistics)Inclusion mapAlgebraMathematical singularityComputabilityDistribution (mathematics)Network topologySocial classMultiplication signQuantificationMusical ensembleTheoryProjective planeDirection (geometry)Different (Kate Ryan album)Connected spaceExtension (kinesiology)Algebraic structureTorsion (mechanics)Algebraische K-TheorieLecture/Conference
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Point (geometry)Abelsche MannigfaltigkeitStability theoryRepresentation theoryExtension (kinesiology)Many-sorted logicChi-squared distributionGroup representationVariety (linguistics)Line (geometry)Multiplication signProjective planeCharacteristic polynomialSampling (statistics)TensorLecture/Conference
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Maß <Mathematik>Elliptic curveAbelsche MannigfaltigkeitTensorCategory of beingGalois-TheorieSymmetric groupField extensionStability theoryOrder (biology)HypothesisLengthExtension (kinesiology)MultilaterationKörper <Algebra>EllipseGroup actionMany-sorted logicL-functionGroup representationStandard deviationTheoremDistribution (mathematics)HeuristicQuantificationNumerical analysisTerm (mathematics)Random matrixMereologyCurveMoment (mathematics)Point (geometry)Analytic setTorsion (mechanics)Chi-squared distributionProof theoryAlgebraic number fieldRange (statistics)Zyklische GruppeMathematicsRight angleQueue (abstract data type)Asymptotic analysisMatrix (mathematics)Connected spaceGoodness of fitMusical ensembleDifferent (Kate Ryan album)SequelAnalytic continuationVariety (linguistics)10 (number)Natural numberFunctional (mathematics)Lecture/Conference
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Position operatorPower (physics)Chi-squared distributionEnergy levelLogicExtension (kinesiology)Stability theoryTheoremSet theoryGroup actionInfinityPopulation densityHeuristicEllipseFrobenius methodFunctional (mathematics)PressureEndomorphismenmonoidKörper <Algebra>L-functionVariety (linguistics)Random matrixFinite setRing (mathematics)Condition numberHypothesisFinitismusOrder (biology)Field extensionRight angle1 (number)QuantificationGenetic programmingMaxima and minimaTotal S.A.Elliptic curveObservational studyModule (mathematics)Multiplication signSquare numberIntegerMusical ensembleGoodness of fitTheory of relativityMatrix (mathematics)VarianceNormal (geometry)19 (number)MorphismusPrime idealCoefficientNichtlineares GleichungssystemSymmetric algebraRational numberSeries (mathematics)Moment (mathematics)Lecture/Conference
29:29
Normal (geometry)Abelsche ErweiterungCurveRandom matrixGroup actionCovering spaceGamma functionOrder (biology)SurfaceMultiplication signDivisor (algebraic geometry)Element (mathematics)Cubic graphNumerical analysisThetafunktionLine (geometry)Field extensionSet theoryRight angleSummierbarkeitMany-sorted logicCycle (graph theory)Symmetric groupAsymptotic analysisPoint (geometry)DivisorCoefficientElliptic curveProjective planeFiber bundleRing (mathematics)Functional (mathematics)CombinatoricsObject (grammar)Category of beingModule (mathematics)Genetic programmingChi-squared distributionPrime idealDifferentiable manifoldHeuristicRational numberAlgebraic curveMathematical singularityFree surfaceINTEGRALExtension (kinesiology)RankingStatisticsKörper <Algebra>Sigma-algebraLecture/Conference
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4 (number)ModulformSquare-free integerPrime idealGamma functionGroup actionEinheitswurzelChi-squared distributionAnalogyIntegerThetafunktionMultiplicationPositional notationField extensionMultiplication signParameter (computer programming)CoefficientNumberSummierbarkeitGauss sumWell-formed formulaRegular graphNumerical analysisGoodness of fitComplex numberProduct (business)Poisson-KlammerElement (mathematics)StatisticsNichtlineares GleichungssystemMany-sorted logicRight angleFrequencyExpressionRational numberL-functionExtension (kinesiology)AlgebraHomomorphismusRing (mathematics)Order (biology)Functional (mathematics)INTEGRALTerm (mathematics)1 (number)Fiber bundleElliptic curveLecture/Conference
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Field extensionCoefficientMoment (mathematics)LogarithmGroup actionMultiplication signIntegerDistribution (mathematics)Genetic programmingFunctional (mathematics)Fraction (mathematics)Equivalence relationChi-squared distributionRootSquare numberGamma functionHypothesisElement (mathematics)Set theoryPoint (geometry)ThetafunktionMeasurementCorrespondence (mathematics)HeuristicRange (statistics)Numerical analysisMaxima and minimaRegular graphLatent heatCondition numberArithmetic meanSine1 (number)SummierbarkeitAbelsche ErweiterungElliptic curveKörper <Algebra>Order (biology)Many-sorted logicUniformer RaumHomomorphismusPermutationTerm (mathematics)Real numberNormal (geometry)Module (mathematics)Sign (mathematics)Power (physics)FunktionalgleichungPhysical lawSheaf (mathematics)Lecture/Conference
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Category of beingDistribution (mathematics)Many-sorted logicAnalytic continuationL-functionRootModulformTournament (medieval)Square numberPoint (geometry)VarianceLimit (category theory)TorusSymmetric algebraDivisorAutomorphismMultiplication signLecture/Conference
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Elliptic curveClosed setStatisticsMultiplication signDistribution (mathematics)Structural loadMany-sorted logicLine (geometry)1 (number)KurtosisLecture/Conference
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Normed vector spaceStatisticsModulformCross-correlationComputabilityThetafunktionCurveMany-sorted logicWell-formed formulaElement (mathematics)CoefficientDirac equationDistribution (mathematics)Elliptic curveAutomorphismLecture/Conference
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Symmetric algebraL-function10 (number)Körper <Algebra>Observational studyIntegerRing (mathematics)VarianceGenetic programmingCoefficientMultiplication signNichtlineares GleichungssystemModule (mathematics)Square numberL-ReiheAlgebraic numberRational numberResultantField extensionFourier seriesSymmetric matrixLogicRight anglePrime idealTheory of relativityNumerical analysisLecture/Conference
Transcript: English(auto-generated)
00:16
Already, half the definitions I'm going to give are there.
00:22
Thank you so much, Sarah. Look, this is, it's been 60 years since I first came to the IAGS, which is pretty much when the IAGS was founded, or at least the first time. Yeah, when it, it was the first year that it moved to Beurre.
00:47
The Residence de l'Ormai just didn't exist, but all the visitors, or most of the visitors and professors, were in Residence Gratia. The air was electric.
01:04
It was, at least for me, it was just extraordinary. There was René Tome understanding topological singularities, structural stability, morphogenesis. And Grothendieck transforming algebraic geometry
01:25
and en passage a good deal of the vocabulary of the way in which we deal with mathematics. Grothendieck would very often like to say, when he talks about something, call it ix.
01:42
He would say, ix n'est rien que igrec. And by that, he would mean there's a change of viewpoint that you have to make, and you have to try to understand. And I was trying eagerly to do that and to learn some mathematics.
02:01
I knew almost no mathematics, actually, when I came first. And I'm so really grateful that I learned so much and sort of felt the inspiration of the IHES
02:22
from its very beginning. I also want to thank Will Hurst for his sort of generous and just overwhelming spirit
02:44
in thinking of various ways of how to put it, of making the mathematical community more congenial, more open, more capable of doing things.
03:02
He is equally generous in the sciences and the arts and the literature. Of course, I don't, and neither does Gretchen, deserve to have our names as named entities
03:25
in this visiting professorship. I'm sort of speechless and don't know what to say about it, but it seems like fun. Anyway, the other thing is this is the fifth lecture today.
03:47
And I can imagine that people are exhausted. We've seen wonderful things. I'm so happy that Sasha Goncharov is the first visiting professor in our named chair.
04:02
And we learned enormous amounts of things in many different directions. So my lecture today, I'll try to be as un-technical as possible.
04:21
And that's pretty easy for my subject because what I'm going to talk about is a project that's an ongoing project with Karl Rubin, which is more experimental mathematics than theoretical mathematics in some sense. It involves a lot of computation. In fact, I will not overload you with computation on a screen,
04:44
but I do have one page, many copies of one page if anyone wants to take a look at it at the end of this lecture. There are various distributions you see here, and I'll try to make some sense of what these distributions are and why we think they're
05:03
sort of vastly interesting. And the other thing I want to say is by making such a lecture, I want to emphasize that we're doing computation, let's say machine.
05:24
Au revoir nature, computation, and neither of us are really professionals. And so if anyone sees something that might be of interest to them, and they have essentially any expertise, they would have more expertise
05:40
than we would, and I'd love to chat with them about it. OK, what do I want to talk about? I want to talk about a variety. V will be a variety over k. k is always a number field here. Let us say a field of finite degree over q.
06:04
And what I'll be interested in is its set of rational points, v of k, k rational points. But I'm interested in it from a kind of a relative standpoint. Let us say I want also to be considering always
06:22
a finite extension field, extension field, l over k, and in some sense the connection between v of k and v of l. Let us say, do we get, I know, say I didn't like nu.
06:43
My mirror is saying nu without introducing extension field. But do we get nu rational points over v of l, given the extension field? So this is a relative question. And by the way, most of the time, but not always,
07:04
I'll be interested in when l over k is Galois. So there's lurking in the initial data. There will be its Galois group. This will be around if l over k is Galois. And I'm going to ask various questions.
07:21
In fact, it's very interesting to sort of change the quantification. You could fix the variety and ask for all l over k's with a given g. You could do various things. And in certain contexts, you could even be more explicit.
07:42
For example, if v is an abelian variety A, you could just forgetting torsion. You could take the Mordell group tensor q and consider the inclusion in the Mordell group of A over l tensor q.
08:04
And in the case where the extension is Galois, you have a g representation, g acts on A of l over k. And instead of asking about new rational points,
08:26
you could be even more specific and choose a representation, say an irreducible representation with a character chi. You can ask, does chi occur in this g representation
08:45
space A of l?
09:01
First of all, to have a certain amount of easy language about this, I want to make a definition. Oops, ah, I see. OK, this is going to go up all the way.
09:34
I should have practiced this before. OK, here's a definition.
09:43
We'll say that v over k is diaphantine stable,
10:03
although abbreviated as ds, for the extension l over k if there are no new rational points.
10:25
Now, there are certain varieties that have very clear characteristics regarding this notion, diaphantine stable. For example, if v is a projective line over k,
10:42
there are no non-trivial extensions that are diaphantine stable for it. So even if v is a variety that contains not only an image of a non-constant image of a projective line or a non-constant image of an open and a projective line, there are none.
11:02
One might ask whether the converse is also true. So there are all sorts of questions you might ask about this. In the case of v to be an abelian variety,
11:26
you might ask, does chi occur in A of l over k? And the standard conjecture would have it that chi occurs in A of l tensor q.
11:47
In the case where chi is an irreducible representation of the Galois group of l over q, the standard conjecture would say that that's true if and only if the Haas of A l
12:01
function of A twisted by chi at the natural point s equals 1 is 0. So this is conjecturally that.
12:24
And after all, we have, therefore, if you want, we can think of this question as an arithmetic question up there and an analytic question down here. And it's fun to sort of try to play one
12:43
of these against the other. So let's take a specific example. How much do you know about the l function analytically? What? How much do you know about the l function? Not much. Not much.
13:01
At s equal 1? Yeah. So this is also the conjecture includes the fact that l of A chi at s equals 1 is defined. That is to say, l of A chi s has an analytic continuation, as Ofer points out.
13:20
So that's part of the conjecture, if you want. Of course, in terms of the quantification here, you could ask, for example, fix a group G.
13:41
You could fix the group G to be the symmetric group s, n and fix the character to be the standard character and ask, for example, does the standard character occur in A of l tensor Q for a fixed abelian variety
14:05
over a fixed field k for infinitely many fields l? That's already an interesting question. And it's also interesting that we can prove this. We can prove it. It's not that hard. We can prove it where A is an elliptic curve.
14:23
And we can almost prove it when A is an abelian variety. That is to say, one can prove the analogous sentence and analogous statement, but at least for n sufficiently large. So there are all sorts of questions that you might phrase from the data of the v over k and l
14:44
over k and the Galois group. So I'm going to now talk about v, an elliptic curve, and k cubed.
15:09
When v is an elliptic curve and k is cubed, the Chantal David and Jack Fernley and Hrzyslowski have pretty interesting conjectures
15:34
about this where G is a cyclic group of order p. Namely, the conjecture is quite simple.
15:48
If p is greater than or equal to 7 for a fixed p, fix it. There are only finitely many.
16:11
L's over q, cyclic of order degree p,
16:23
that are a diaphantine unstable. That is to say that you get actual extension of points when you pass from k to q. In other words, all but finitely many are diaphantine stable, the conjecture in this range.
16:41
And the way they do it is by using a rather beautiful but curious use of random matrix heuristics.
17:01
And the random matrix heuristics will also give asymptotics. The answer is false for p equals 2, 3, and 5. But they give asymptotics for p equals 2, 3, and 5. And I might return to this a bit later. So it's false, but with asymptotics.
17:25
And proved false or conjectured false? Proved false. Proved false. Proved false for the thing is I'm worried about being over q.
17:51
But if you allow me to make a slight base change, it would be proved false for any of them, 2, 3, and 5. 5 is really interesting.
18:01
It involves a certain curve called Brings curve. I don't know whether anybody's heard about it. We were. Does that mean the conjecture will work over another number field or not? Will the conjecture work over another number field with other p's? In other words, if you change the number field,
18:22
you change the 7. The heuristics is about the L function or about the? Effectively, the heuristic is both about the L function. Random matrix heuristics is about the L function. And I should say also that Ruben and I, Carl Ruben and I,
18:43
have heuristics as well. But we call them naive heuristics. They're dependent only on certain distributions having a property that we think they have
19:00
in various pieces of data that we've collected. So if you know it, the tensor in this q, that is the stability from the L functions, then is it easy to get back to without tensor with q? So you know that tensor with q doesn't change.
19:22
But you define stable in that. Oh, tensor with q. Yeah, no. I mean, the thing is, if you have an elliptic curve over q, the only difference, that's a good question. The answer is yes. But that's a good question. Because you can bound the torsion.
19:40
Exactly. Well, you certainly can bound the torsion. But you can do even better. Great. OK. So our heuristics, which I am going to give you in a moment. But before I do that, let me tell you a theorem that we proved a number of years ago.
20:03
It's sort of just published in the American Journal, which got us really interested in trying to understand this whole thing in some serious way. So here's the theorem. This is with Ruben.
20:23
I know you're supposed to put a dash there, right? I'm going to do me. OK. Ruben and me. The theorem is, let v be one of two things.
20:41
It could be a curve, geometrically irreducible, of genus greater than 0.
21:02
Or it could be an abelian variety, geometrically simple. That geometrically simple actually is important. And v is over a field k.
21:22
And there'll be a hypothesis on k, which I'll tell you a bit later. It's a mild hypothesis. I don't want to, OK. And so here's the theorem. And the theorem is loved by mathematical logicians because there are lots of quantifiers.
21:41
There exists a set of primes, p, of positive density, positive density, so fixed p, such that for all integers,
22:14
n equals 1, 2, 3, and so on,
22:21
fix now p to the n. Oh, no, forget that fixed p. Yeah. But fix that one. Such that for any fixed p to the n, there are infinitely many, what?
22:45
Fix means for every n. Oh, you choose any p in this set and choose any n in this set. And then for any pn, you have that. Yeah, that's what I mean. For every npn. OK. That's right. I could see, yeah.
23:00
You don't need, if you try to quantify it, don't say. Right, right, right, right, right, right. OK, very good, very good. You're right. I just wanted to make sure I was not, that the infinitely many wasn't confusing. Anyway, there exists infinitely many extensions, cyclic Galois extensions, L over k, of degree p to the n.
23:42
So let's say for fixed p to the n, there are infinitely many Galois extensions. Maybe I should have said that. L over k that are diaphantine stable for v over k.
24:02
Oh, did I? So there are some various things I have to say. I have to tell you what the hypothesis on k is, which I'll do in a moment. A set of positive density, it's what
24:23
we might call a Chebertov set. There's some finite extension, and you put conditions on these p's having to do with Frobenius for things lying over those p's. And you get positive density. We expect that this is of density 1.
24:42
And in fact, we expect even better than that. Infinitely many, this is also rather an impoverished infinitely in the sense that if you go up to a conductor where you have gotten x cyclic Galois
25:02
extensions of that degree, it'll be on the level of x over log to some power of x that we actually prove exists. So we don't even get a positive density of such cyclic Galois extensions. But we expect. Could you say it's Chebertov set, but if it is density 1,
25:25
it means complement of a finite set. Yeah, yeah. You expect that? Oh, I expect complement to a finite set, exactly. OK. Exactly. The hypothesis on k is simply, I won't write it, but I want the Jacobian of c to have its endomorphisms.
25:47
The endomorphism ring of the Jacobian of c I want to be all defined over k. Let us say the endomorphism ring over k bar of the Jacobian of c is the same as the endomorphism ring of k over k.
26:01
And with a, it's pretty much the same thing. All endomorphisms of a over k bar should be defined over k. So if k doesn't satisfy this hypothesis, you have a single variety in question. You just pass from k to whatever field
26:24
you need to make every endomorphism rational, and you get this theorem. So the next thing I would like to do is go back to the question, the more general question,
26:41
does chi occur? And is it connected? And do it in a manner that's connected to the Haase-Weil function. But we're going to do it for a now being not only a Boolean variety, but an elliptic curve, and k being q.
27:20
Well, perhaps I could tell you I was going to end with the conjecture that we have. But perhaps I'll tell you the conjecture that we have, which is, in some sense, inspired by the David-Fernley-Kaczylowski conjecture, but is motivated by what we call our naive
27:44
heuristic rather than a random matrix heuristic. We conjecture that if we have e over q,
28:02
and if we consider a billion Galois characters, let us say, or Dirichlet characters, if you wish, chi from the Galois group of l over q to c star.
28:20
And you consider the set x of all such chi's such that, and now, I don't need to use the conjecture that the l function extends because it's gone to, all such chi's such that le chi 1 equals 0.
28:44
But I exclude the chi's that have the order of chi. I exclude the ones that are of order greater than or equal to 5. And I also want it not to be order 8, 10, and 12.
29:11
And our conjecture is this set x is finite. But I'll try to give you some sense of how we come to it.
29:27
And for that, I'm going to move to the combinatorial way of thinking of l functions,
29:40
namely, theta elements. Or maybe one thing I might do is, this is a digression. And if I have time, I might have time.
30:00
Our conjecture that we give is more precise than that because it gives asymptotics for the chi's that are missing there. And those asymptotics compare well with the asymptotics of David, Ferley, and Kacilevski that they get from random matrix heuristics.
30:23
But there are kind of wonderful things you can do for the missing primes, for example, p equals 2, 3, and 5. And perhaps I should, since I mentioned my conjecture, talk about 3. I'll try to do it fast.
30:41
So the question for 3 is, how many cubic cyclic extensors of Q have the property that a given elliptic, you fix your elliptic curve over Q, how many cubic cyclic extensions are there where the elliptic curve picks up more points over that cubic cyclic extension?
31:03
What in the sense of the rank? More in the sense of the rank. As you'll see, it's not going to matter because it's going to be governed by a pencil of cubic extensions whose total space is a curve of a genus greater than 1. And so we can use Manning, Mumford, and faultings
31:23
to show that you have a whole pencil of rational points over the p1. And for each rational point, you get anyway. So I like this example. I'm going to try to do it fast so it doesn't disturb the rest of my talk too much.
31:42
But the easiest way to try to understand this is you take E cross E cross E, the elliptic curve that you're looking for, cubic cyclic extensions for which it's not diaphantine stable. The symmetric group of order 3 acts on this.
32:01
And so you can divide by 3 cycle. And this is your very generous cubic cyclic extension that you're going to milk. You can, if you want, sum to 0.
32:23
Right, you can take the sums. And you can take 0 in here and take the inverse image of 0 in here. So it's the set of cubic things that sum to 0. That would give you a surface rather than a 3-fold.
32:44
And you could go all the way. You could divide by the symmetric group of order 3, which gives you sym 3 of E, which also sums to E. And if you take the inverse image of 0 there, you get something that I'll just call p. It's really a projective plane.
33:01
It's in some sense can be viewed as the dual projective plane of the elliptic curve. This is a double cover. And it will be ramified at a curve of degree 6 with 9 singular points.
33:20
It's the dual curve to the elliptic curve. And it's a kind of k3 surface wannabe. This x is, if you blow up those nine points, you get 18 projective lines. And so this x is k3 surface of Picard number 19,
33:42
18 plus the ample divisor. And so every time you find a rational curve in this, you just pull it back. So let's suppose I find the rational curve and take the pullback of that rational curve. I got a curve in here, which, well, the pullback,
34:06
I want the normalization of the pullback. So this is a smooth curve now, which maps to E cross E cross E. And here we are. We have a pencil. This is a cubic cyclic extension. We have a pencil of cubic cyclic candidates.
34:22
We take any rational point there, go up there, and project to, say, the first factor. And you get a cubic cyclic point in E. And as I said before, Monning, Mumford, and Foltings tells you you actually get infinitely many L over k. The C is the genus greater than 1.
34:42
That's the three. The five is, as I say, even more interesting. But for a lack of time, I'll go on to the analytic story, which is this.
35:03
But I'm going to restrict it to elliptic curve over k equals q. OK, now there's a whole essentially mini subject
35:22
that connects the values le chi 1 to more combinatorial objects, like modular symbols, as it was mentioned in Sasha's talk in the beginning of the hour. And lots is known about modular symbols.
35:44
In particular, lots is known about the statistics of the values of modular symbols. But what I want to pass to is not modular symbols, but things that are built out of modular symbols that we call theta elements.
36:05
And they depend upon a field extension, L over q, a cyclic Galois, of course. It's cyclic field extension, say of degree, call its degree d.
36:23
And I'll call it just theta sub L. And the thing about theta sub L is it's really more an arithmetic object than an analytic object that lives in the integral group ring of the Galois group of L over q.
36:47
So when I write it out, I'll write it as sigma for gamma in that Galois group of some coefficient CL gamma times the element gamma in the group ring.
37:05
So these guys we'll call theta coefficients. And the virtue of this theta L is that if you take any chi from that
37:24
CL gamma is in what? What? In z? In z. CL gamma? Is in z. Is in z. Oh, yeah, yeah, yeah. These theta coefficients are in z. So they're integers.
37:42
And if you have any chi from the Galois group of L over q to c star, the virtue of this theta is the following. You can apply chi not only to the group, you can apply it to the group ring as a homomorphism
38:00
to c of algebras. When you apply it to the group ring, you get chi of theta L is equal to something which is visibly non-trivial, non-zero, non-zero. And more elementary, well, it involves a period and a Gauss sum maybe. But it's just non-zero times L e chi 1.
38:29
So if we're interested in is L e chi 1 0 or not, we're really asking is, if I replace this a by e,
38:40
we will be asking is chi of theta L 0 or not. And these are equivalent, not by conjecture, but they are equivalent.
39:10
Oh, I don't want that. Let me get rid of this.
39:30
Well, just to give you a sense of what's going on, like my L over Q is of any degree d, now let's suppose that the degree of L over Q
39:45
is a prime, although everything I say here with more argument and notation has qualitative analogs
40:01
for any d. OK, if d is a prime, take a look at that chi of theta L. Chi of theta L is a sum of integers cl gamma,
40:21
chi's of gamma, and these guys are p-th roots of unity. And so this is in z bracket zeta p.
40:40
So you learn that, hey, we have a certain element in, a cyclotomic element, if you wish, element in z bracket zeta p. We're asking is it 0 or not. And the answer is 0 if and only if all the c's are equal.
41:05
So chi of theta L is 0 if and only if all of the cl gammas are equal to cl gamma primes for all gamma and gamma prime in the Galois group.
41:27
OK, so this is the sort of thing that might get you interested in asking statistics about these numerical values. I mean, how often are they equal, all of them
41:41
equal for a given theta element? So for example, before you do any statistics,
42:00
you should try to figure out what regularities these cl gammas have. So we want to understand for every L over q cyclic Galois extension of degree p. And for the gammas in their Galois group,
42:23
we want to understand these integers cl gamma. And already, we're specifically interested in whether what happens when they're all equal. Well, there's some regularities. Before you do any statistics, you better take care of the regularities.
42:45
And the first is the sum. One has a very clear expression for the sum of all the cl gammas. Let us say, in the case where I take m to be the conductor of L,
43:03
and at least to say what I'm saying without the extra terms, I'm going to assume that this is square free and prime to p. If you take the sum of all these cl gammas,
43:28
you get the following interesting thing. It's the product for all primes dividing the square free number m of the L Fourier
43:40
coefficient of the modular form for the elliptic curve. I'll call that Al minus 2 times some number, a rational number. And that rational number depends on nothing except for e. Well, in fact, this rational number is a non-trivial multiple of the L function of e over q
44:05
at 1. And so therefore, this is 0 if the L function over q of e at 1 is 0 and non-zero if not. In any case, we see that in order to get such an effect, perhaps I'll
44:27
put it again on this board, this is going to happen, at least in the context where I make the hypothesis of square free prime to p, that every cl gamma, all of them,
44:41
have to be equal to 1 over p times, in the sense that we'll call it the right hand side of that equation. What is that Al minus 2? What? In the equation, you want photos of A? Al minus 2, where L runs through all the primes
45:03
dividing m. And Al is? Al, oh, yeah, good question. In fact, I want one more requirement with its prime to p and to prime 2, the conductor of e.
45:21
And Al is the usual thing. Usual thing, yeah, yeah, yeah. And there's a similar formula. It's more complicated in general. So under these hypotheses, we get not only for chi of L to be 0, let us say, for us to get a 0 as a.
45:43
Al is the trace of Riemannian or minus trace of Riemannian? It's the trace of Riemannian. Yeah. So we get this is equivalent to that. So that does suggest that maybe one
46:01
should take a look at the distributions related to these values of these theta coefficients. I mean, after all, for fields of degree p, in order to get a 0, you have to have every one of these theta coefficients
46:21
to be equal to this specific number. And so what's going on here? So the natural thing to do is to produce data. I'll consider it data.
46:42
And by the way, I'm going to do this data not only for d. Oh, I should also give you one more regularity before I give you this data. Namely, the Atkin-Laner, or what
47:05
I'll call the functional equation, identifies essentially one theta coefficient with another in the following way. Especially, I'm still under the hypothesis
47:20
that m is prime to p conductor of e and square free, in which case the identification is fairly simple. Namely, if I take the Galois group of L over q, there is an involution. I'll call it I. It's a permutation of order 2.
47:45
It's not a homomorphism. Needn't preserve 0. But there's an involution, which gives a condition on the values of the CL gammas. Let us say, if I consider CL and apply the involution
48:02
to gamma, it's equal to pretty much CL gamma up to a sine. And the sine is the root number of the elliptic curve. So if we're trying to make data, which I was about ready to do on this board, I don't necessarily care to take all of the CL gammas.
48:23
I will. But I should understand that there is a kind of correspondence between them in pairs. So here's what I want to do. I want to take all the data.
48:41
I'll call it data of e and d. And this is just the set of all integers for the moment. I'll normalize them to make good use of them. Where L over q runs through all cyclic extensions of degree d.
49:05
And if d is even, I want it to be real cyclic. And I want gamma to range through all elements in g which are not equal to their image under that involution.
49:24
These we call generic, since the involution has a maximum of two fixed points and sometimes none. And that's my data. But if that were my data and I asked, what's the distribution that's determined by them,
49:42
it would just flatten out to something horrible. So I have to renormalize it. And this is how I renormalize. CL gamma. I multiply by the square root of d. And I divide by the Euler-Fee function of the conductor of L
50:02
times log of the conductor of the square root of log of the conductor. So that's a whole lot of data. And the question is, does this converge to a distribution? If it does, what sort of distribution
50:21
does it converge to? And how might it connect with the issue of how often is CL gamma going to be equal to that specific thing? So here's our conjecture. So we begin.
50:40
You're now on a computer. At this point, it's a total experiment. But there is some heuristics why you put those factors. What? There is some reason you put those factors. Yeah, if you want to know the CL, I will tell you just very rapidly. The CL gammas are really a sum of phi of m modulus symbols. The modulus symbols move up into modulus symbols
51:05
with denominator m. The modulus symbols move up by log m, by roughly log m, or they're bounded by a constant times log m in terms of the denominator. So in order to bring this to a reasonable range,
51:22
I have to divide by this. And I multiply by this because I want uniformity for all d. And that's OK. So that gives you some rapid reason why I do this normalization. But anyway, here's our conjecture.
51:49
This converges. So the condition is fixed e and fixed d. I fix e, I fix d. So the data of ed is this whole set.
52:03
And I vary l through all real cyclic extension of the degree d. And I vary gamma through all, as I call, generic elements of the gamma group of l over q. And by the way, the other ones I've avoided, the ones the gamma is equal to i gamma, we call special.
52:22
And they produce different distributions. But anyway, I'm going to give you one. And we can discuss how that distribution relates to the special ones. OK, this, meaning that converges to a distribution,
52:41
which we call lambda ed of t. And you know what that means. That means if I integrate this thing, if I think of it as a function or a distribution, if I integrate this thing
53:03
over t in some range, that would give me the percentage of elements in this set that live in that range. Do somebody? The measure in every point you have a dt?
53:21
Yeah, yeah. Well, I'm thinking of it as you see. Yeah, it'll be multiplied by dt. That would be the measure, exactly. But the distribution has the following properties. One, it's continuous. It's also, we have a sort of clear conjecture of its shape.
53:44
But just continuous is enough, except possibly at t equals 0. Two, if d is large enough, it is even continuous there,
54:08
simply continuous. Three, the limit as d goes to infinity, which is the reason for the square root of d there,
54:22
to deal with this, of these lambda dt's edt's is Gaussian
54:41
with variance equal to some elementary term, it looks like a bunch of Euler factors, times the L function of the symmetric square of the automorphic form, modular form attached
55:03
to f at the point s equals 2. So that's our conjecture. And if anybody is interested in wanting to take a close look at one, we've done loads of these.
55:20
And I didn't want to overload the lecture with screen after screen of data. But I have an example here of one page. And if anybody is interested, they could take a look at it. I have many copies. We consider the only elliptic curve that people start their empirical journeys with.
55:45
And Sarah can guess which one. 11? Yeah. There are only two, you see. Either it's 11 or it's 37. I mean, one or the other. 11 is better.
56:01
We did 11 first. And so we did d equals 3. And the distribution is sort of spiky. It's always spiky. So by the statistical jargon, the jargon of the statisticians, there's kurtosis. That is, it's spikier than Gaussian.
56:22
And we don't know why. Anyway, it's this big. For d equals 7, it's this big. For d equals 19, it's next. And by the time you get to d of 101, it's so close to that dotted line, which is Gaussian, that it's hard to imagine it's not moving towards Gaussian.
56:40
And that happens with all the elliptic curves we've tried. And so we became fascinated by these things. We know no statistics. When I call, I have a friend who is a statistician. I frantically call her every once in a while about some statistical thing.
57:01
And she will always say, Barry, read a book on statistics. So anyway, so I'm reading books on statistics. Anyway, whatever, we've become fascinated by these distributions, which, in principle,
57:25
doesn't have anything to do necessarily with automorphic forms on GL2 or any automorphic. You can always twist by Dirichlet characters and see what you get. And there are these statistics.
57:41
And they seem to depend somewhat. In fact, visually, in our computations, they do depend a bit on the elliptic curve, therefore on the automorphic form. So they're interesting. And we would love to know, let's say, a closed formula for them.
58:01
We haven't found, nor do we have any guesses. In any case, what this does, wait, I have two more minutes. What do I have? What? OK, anyway, here I go. I'm just going to tell you our conjecture that follows,
58:22
we think, from this and from also a conjecture about the correlations of theta elements of the given theta, of theta coefficients of given theta element. So our conjecture is, let e be an elliptic curve over q.
58:49
Let m over q be any abelian field of rational numbers. I'd say the interesting thing here
59:00
would be of algebraic numbers. But it contains only finitely many subfields
59:28
of degree 2, 3, or 5.
59:47
Then e of m is finitely generated.
01:00:03
I should say that this is actually true. This would be true thanks to Cotto, Ribet, and Rorlich if M, for example, simply was unramified outside finitely
01:00:21
many primes. But here, we're not assuming that it's unramified outside finitely many primes. There are many abelian fields that are pretty interesting that are contained in this. And one of the reasons why it might be interesting
01:00:40
is that every time you have such a field, such an M, where E of M is shown to be finitely generated, Hilbert's 10th problem is false. So this is connected to some recent studies of Hilbert's 10th problem, where people
01:01:03
don't know about the total, the maximal abelian field. They don't know. Is it true or false? And the logicians don't know what to believe there. Which one will be there? What? Which one will be there?
01:01:20
10th. That the ring of integers in that field would be what's called diophanting. That is, there's an algorithm that determines yes or no.
01:01:40
Does an equation with coefficients in that field have a solution in the ring of integers of that field have a solution in the ring of integers of that field? OK. I think I've forgotten. I think this is my, yeah, OK. Thank you. OK, are there any questions?
01:02:10
Why symmetric square to have appears? You're right. Why is the, oh, why does the symmetric square? Oh, no, that's a good question. The reason is that, I mean, we conjectured this.
01:02:26
And we conjectured this even before the data. Because if you think of the symmetric square and you think of the Fourier coefficients,
01:02:41
they'll be squares, a of p squares. And those Dirichlet series control the variance of modular symbols. And this is connected to modular symbols.
01:03:03
Oh, and if you average such things, not for the theta coefficients, but for the modular symbols, you average it over all modular symbols and normalize appropriately and ask for the appropriate variance. The variance is proved to be L function
01:03:20
of the symmetric square. It's AP square. You have to think of it that way. You did not elaborate about the relation with the analytic results. But you mentioned Rolleish. If you twist by a character of the p and you vary the character, the results of Rolleish,
01:03:41
the more t, you have no zero, is that right? Yeah, yeah. So how does it connect? It connects perfectly with it, yeah, yeah. But they're always in a specific situation where there are only finitely many ramified primes.
01:04:02
Well, that's the case. I think that's, yeah, that's the case. So what they prove, they prove this conjecture if, forget this, they prove this conjecture if m over q is unramified outside finitely many primes. Oh, I see. Yeah, OK, thank you.