User vesselin dimitrov - MathOverflow

The elliptic Lehmer problem for several independent algebraic points

2013-03-05T19:00:35Z

The higher dimensional Lehmer problem asserts that if $\alpha_1,\ldots,\alpha_r$ are multiplicatively independent non-zero algebraic numbers generating an extension of $\mathbb{Q}$ of degree $d$, then $h(\alpha_1) \cdots h(\alpha_r) \geq c(r)/d$. This arose in the work of Amoroso and David, who have shown this up to a logarithmic factor, in a generalization of Dobrowolski's bound. As a corollary, Amoroso and David obtain the truth of the original Lehmer conjecture under the assumption that $\alpha$ generates a normal extension of $\mathbb{Q}$.

The natural analog for the case of elliptic curves ought to be:

If $P_1,\ldots,P_r \in E(\bar{\mathbb{Q}})$ are independent algebraic points generating a number field of degree $d$, then the product of their canonical heights is at least $c(E,r)/d$.

Still, I have not yet seen this statement explicitly mentioned in the literature. (Is it indeed expected to hold?)

Usually, results on the original Lehmer problem transfer without great difficulties over to the case of CM elliptic curves (essentially because the CM hypothesis allows to lift Frobenii). For instance, the literal analog of Dobrowolski's bound for CM elliptic curves is due to Laurent, from 1981.

Question. In light of this, has the above statement about $r$ independent algebraic points on a CM elliptic curve been proved up to a logarithmic factor? In particular, is the Lehmer conjecture known to be true for points on CM elliptic curves that generate a normal extension of $\mathbb{Q}$? Are there any results available in the literature?

The critical exponent in the multiplicative order of 2 modulo primes

2013-03-02T02:00:47Z

This is a sequel to this MO question:

http://mathoverflow.net/questions/60441/the-multiplicative-order-of-2-modulo-primes

As shown in Charles Matthews' paper linked to there, it is not hard to show that for each $\delta > 0$ there is a $c = c(\delta) > 0$ such that the set of primes $p$ for which the multiplicative order $n_p := \mathrm{ord}_p(2)$ of $2$ modulo $p$ satisfies $n_p < c\sqrt{p}$, has density $< \delta$. In particular, the set of primes with $n_p < p^{\frac{1}{2} - \epsilon}$ have zero density.

My question is, is $1/2$ really the critical exponent? For $\epsilon > 0$, is there a positive density of primes $p$ with $n_p < p^{\frac{1}{2} + \epsilon}$. Moreover, is there a $C < \infty$ for which the set of $p$ with $n_p < C\sqrt{p}$ has positive density?

I would also like to ask about the analogous question for elliptic curves: is $1/3$ really the critical exponent there? Given a point $P$ of infinite order, is there a $C < \infty$ for which the set of $p$ such that the order of $P \mod{p}$ is $< C \sqrt[3]{p}$ has positive density?

Answer by Vesselin Dimitrov for Varieties where every non-zero effective divisor is ample

2013-03-02T07:53:13Z

It is a general fact that on any simple abelian variety, an effective divisor is ample. The following result underlies the usual algebraic proof of the projectivity of abelian varieties (defined initially only as complete group varieties). It is therefore rather standard; the first reference which comes to mind is Lemma 8.5.6 on page 253 in the abelian varieties chapter of the book Heights in diophantine geometry by Bombieri and Gubler, from which I quote literally.

Let A be an abelian variety and $D$ an effective divisor such that the subgroup $$ Z_D : \hspace{3cm} { a \in A \mid a + D = D } $$ is finite. Then $D$ is ample on $A$.

If the abelian variety $A$ is simple, and $D$ is non-zero, then the $Z_D$ is a fortiori finite, since it is a proper algebraic subgroup of $A$; and it follows from the quoted Lemma 8.5.6 that $D$ is ample.

An application. A surjective morphism $f: A \to X$ from a simple abelian variety onto a positive-dimensional projective variety $X$ is finite.

Proof. Choose $H \subset X$ an ample divisor. The divisor $f^*H$ is effective on $A$, hence it is ample. This is equivalent to $f$ being finite.

Points of minimum Arakelov height and harmonic arithmetical varieties

2013-02-09T14:25:10Z

Added. (28/2) To put it less pompously (and more vaguely, less concretely), I wanted to relate the impression that it is the general rule that an Arakelov (i.e., geometric) height on an arithmetical variety has an isolated minimum. For example, Zagier has shown (Algebraic numbers close to both $0$ and $1$) that for the subvariety $Z : 1+x+y =0$ of the linear torus $\mathbb{G}_m^2$, the minimum, away from the few torsion points on $Z$, of the standard Weil height is $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big)$, with equality if and only if $x$ or $y$ is a primitive $10$th root of unity, and this minimum is isolated.

Another example, though of somewhat different flavor: the minimum height of a totally real algebraic number is, again, $\frac{1}{2} \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.2406059\ldots$, and this minimum is isolated. The lim inf of the height of a totally real algebraic number is at most $0.2732831\ldots$, and indeed it has been suggested that this is the lowest possible accumulation point for heights of totally real algebraic numbers. One realizes this accumulation point by taking, iteratively, $\xi_0 := 1$ and $\xi_n - \xi_n^{-1} := \xi_{n-1}$.

I wonder whether it is a general feature of both (a) Arakelov heights on arithmetical varieties (apart from the obvious examples); and (b) totally real or totally $p$-adic points on semiabelian varieties, to have an isolated minimum for the heights of their algebraic points.

Original post. By a polarized arithmetical variety I will mean a pair $(X,L)$ of a finite-type proper regular integral scheme $X$ flat and generically smooth over $\mathbb{Z}$, and a relatively ample invertible sheaf $L \in \mathrm{PIC}(X)$ equipped with an $F_{\infty}$-invariant hermitian metric $\| \cdot\|$ on the associated holomorphic line bundles $L_{\mathbb{C}}$, such that $\|\cdot\|$ is the uniform limit of positive $C^{\infty}$ metrics.

There is then an Arakelov height function $h_L$ on the algebraic points $X(\bar{\mathbb{Q}})$, given by the arithmetic intersection number of the associated multisection with $\hat{c}_1(L)$, divided by $[\mathbb{Q}(x):\mathbb{Q}]$. Let me call such a polarization $(X,L)$ strict harmonic if the set $$ { x \in X(\bar{\mathbb{Q}}) \quad | \quad h_L(x) = \inf_{X(\bar{\mathbb{Q}})} h_L } $$ is Zariski-dense. Let me call $(X,L)$ harmonic (or non-strict harmonic) if $$ \inf_{X(\bar{\mathbb{Q}})} h_L = \liminf_{X(\bar{\mathbb{Q}})} h_L, $$ where the lim inf is under the Zariski topology. One may ask whether or not the two conditions are in fact equivalent.

One can show, as a consequence of the arithmetic Riemann-Roch theorem, that if $(X,L)$ is harmonic, the infimum equals the normalized arithmetic self-intersection (or arithmetic volume) $L^{\dim{X}} \Big/ L_{\mathbb{Q}}^{\dim{X_{\mathbb{Q}}}} \cdot \dim{X}$, and that moreover, the points of minimum height have their Galois orbits equidistributed in $c_1(L)$ (which, by definition, is a uniform limit of Chern forms of smooth metrics). Examples of strict harmonic arithmetical varieties include the canonical symmetric polarizations of abelian schemes over the full ring of integers of a number field (in which case the height $h_L$ is just the Neron-Tate canonical height, and the points of minimum height are precisely the torsion points); and, on the other hand, projective space with its standard Weil height, or more generally, with the canonical heights of Call-Silverman. A setup which generalizes both these examples is to have a self-map $f : X \to X$, an isomorphism $f^*L \cong L^{\otimes q}$ (over $\mathbb{Z}$, not just generically!) with some $q > 1$, and the height $\hat{h}_f(x) := \lim q^{-n}h(f^nx)$. This is an honest Arakelov (i.e., geometric) height precisely when both $f$ and the isomorphism $f^*L \cong L^{\otimes q}$ are defined over globally over $\mathbb{Z}$, rather than just generically over $\mathbb{Q}$.

Two questions:

Is it true that semistable elliptic curves over $\mathbb{Q}$ (this case being the simplest), or more generally abelian varieties with non-integral moduli, are never harmonic in the above sense, with respect to any symmetric canonical polarization? with respect to any (ample, symmetric) polarization?
Is it true that an arithmetical surface of genus $> 1$ is never harmonic in the above sense? In particular, is the canonical polarization $\omega = \omega_{X/\mathbb{Z}}$ ever harmonic?

Remark on question 1. For an elliptic curve $E/\mathbb{Q}$ with minimal discriminant $\Delta$ and the canonical polarization with $L := \mathcal{O}_E([O])$ -- note that this polarization involves the canonical compactification of the Neron model, as well as the canonical metric on $L$ from Arakelov theory -- everything reduces to the question of whether there exists a sequence of points with Arakelov height $h_L(x)$ converging to $-\log{|\Delta|}/24$.

Obviously, I am interested in this broader question: what are the arithmetical varieties for which admit a generic sequence of points of minimum height?

Is the canonical height of a totally p-adic point on an abelian variety bounded away from zero?

2013-02-27T12:33:03Z

Inspired by the result of Schinzel and Smyth that a totally real number other than $0$ and $\pm 1$ has height at least $\frac{1}{2}\log \Big( \frac{1+\sqrt{5}}{2} \Big) = 0.240659\ldots$, Bombieri and Zannier discovered that totally $p$-adic algebraic numbers which are not roots of $1$ likewise have height bounded away from $0$. More precisely, the proved that the lim inf of the height over the totally p-adic algebraic numbers is between $\frac{1}{2}\frac{\log{p}}{p+1}$ and $\frac{\log{p}}{p-1}$; cf. Ch. 4.6 of the Bombieri-Gubler book (Heights in diophantine geometry). Both results (real and $p$-adic) can be explained as consequences of a Galois equidistribution property for algebraic numbers of small height.

Similarly, for the totally real points of an abelian variety, the Archimedean Galois equidistribution of Szpiro-Ullmo-Zhang implies (S. W. Zhang, Equidistribution of small points on abelian varieties, Corollary 2) that the height of a totally real (non-torsion) point on an abelian variety is bounded away from $0$.

Now, there are various non-Archimedean equidistribution results in the literature in the spirit of the Szpiro-Ullmo-Zhang theorem. Does any of them imply a lower bound on the height of a totally $p$-adic non-torsion point on an abelian variety? Is there anything otherwise published on this problem?

Added a little later. By W. Gubler's "tropical equidistribution theorem" (which is really a statement about the equidistribution of $p$-adic valuations mod 1), we do know that the answer is positive for abelian varieties which do not have potentially good reduction at some place above $p$ (that is to say, they acquire some $\mathbb{G}_m$-part at some place above $p$).

But what about the proper case?

An example. The level $N$ division field $K_N := \mathbb{Q}(A[N])$ of the abelian variety contains the level $N$ cyclotomic field $C_N = \mathbb{Q}(\mu_N)$, so a prime which splits completely in $K_N$ is congruent to $1 \mod{N}$, hence $> N$. Hence, for a given $p$, the abelian variety contains only finitely many totally p-adic torsion points.

The torsion point count in higher dimension

2013-02-08T14:33:43Z

It is an easy consequence of the Serre open image theorem that for the torsion point count on elliptic curves, the following possibilities arise.

If $E/\bar{\mathbb{Q}}$ is an elliptic curve without CM, then the number of torsion points $x \in E$ with $[\mathbb{Q}(x):\mathbb{Q}] \leq d$ is $\asymp d^{3/2}$ as $d \to \infty$.
If $E/\bar{\mathbb{Q}}$ is an elliptic curve with CM, then this number is $\asymp d^2$ as $d \to \infty$.

This appears on page 44 of Serre's book, Lectures on Mordell-Weil. As stated there, it is easy to show that, for a $g$-dimensional abelian variety, the corresponding count is bounded by $\leq O(d^{N})$ with $N = N(g) < \infty$.

Should we expect, for $g$-dimensional abelian varieties, the count to be always asymptotic to $d^{\alpha}$ for a finite set of exponents $\alpha$ depending on $g$? Are there any clues as to the spectrum of those exponents?

Second, I wanted to ask about any results on the uniform torsion count problem giving bounds that are uniform in $E$ but still polynomial in $d$. (Thus I do not ask about Merel's uniform boundedness theorem and its relatives). For example, restricting to $g$-dimensional abelian varieties over $\bar{\mathbb{Z}}$ (that is, with integral moduli, or with everywhere potentially good reduction), or at least to the abelian schemes over the spectra of rings of integers of number fields of bounded degree, it is clear a priori that there is a uniform exponential bound in $d$. Should we expect this uniform bound to be strengthened to a polynomial bound, or even (for elliptic curves) to $Cd^{2+\epsilon}$? What about the generalization where we count the small algebraic points, say those of (canonical) height $< 1/d$?

The height of an orbit under rational self-maps

2013-01-30T13:32:04Z

I have this basic question on which, strangely enough, the algebraic dynamics literature appears to be silent. But the question does not appear to be totally trivial or uninteresting to me - am I wrong?

Question. Let $\varphi : \mathbb{A}^r \to \mathbb{A}^r$ be a regular map defined over $\bar{\mathbb{Q}}$, and let $x_0 \in \mathbb{A}^r(\bar{{\mathbb{Q}}})$ be an algebraic point. What can be said about the growth, in $n$, of the (logarithmic) height of the iterates $h(\varphi^n(x_0))$? More generally: the same question with an endomorphism $\varphi : X \to X$ of any quasi-projective variety.

For example, one can obviously realize the growth rates $O(1)$ (iff the orbit is pre-periodic), $O(\log{n})$, $O(n)$, and $O(d^n)$ for all $d \in \mathbb{N}$.

The question is of course trivial for $r = 1$, or more generally if $\varphi$ extends to a morphism $\mathbb{P}^r \to \mathbb{P}^r$. Or more generally still, for $\varphi : X \to X$ with $X$ projective. (In this case, only the mentioned growth rates are possible).

One (e.g., I) can characterize the $\varphi$ with $h(\varphi^n(x_0))$ having a small growth rate (e.g., bounded by $O(n^{1/r})$), and I wondered whether this is of any interest, or completely trivial.

EDIT: More generally, consider rational self-maps $\varphi : X \dashrightarrow X$ of a projective variety $X$ over $\overline{\mathbb{Q}}$, and a point $x_0 \in X(\bar{\mathbb{Q}})$ whose orbit is contained in the domain of $\varphi$. Then I can show, for instance (is this self-evident?) that if $h(\varphi^n(x_0)) = o(\log{n})$, then $x_0$ is pre-periodic.

NEW EDIT (2/17): On returning to this question, I realized just now that the statement from the previous edit (from 1/30), as written, was indeed a trivial consequence of the rational point count and the pigeonhole principle, the latter forcing the characterization of pre-periodic points as above, with the $o(\log{n})$ improved by, roughly, $\frac{1}{\dim{X}}\log{n}$. Sorry about that. What I really wanted to say was not $o(\log{n})$, but (essentially) $\leq \log{n}$. In other words, the factor $1/\dim{X}$ in the trivial lower bound may be improved, in the setup of the previous edit, to $1$: more precisely, if $\log{n} - h(\varphi^n(x_0)) \to +\infty$, then $x_0$ is pre-periodic. It is this that I intended in my remark that the logarithm is the slowest growth rate of a non-preperiodic orbit. (Note that $h$ is the logarithmic height; thus, for a non-zero translation of $\mathbb{A}^1$, the height of the orbit is just $\log{n} + O(1)$.)

In fact, excluding certain basic, well understood cases, of which translations of $\mathbb{A}^1$ are the prototypical example, and in all of which the height is asymptotic to $d \log{n}$ for some $d \in \mathbb{N}$, the trivial lower bound $\log{(n^{1/\dim{X}})}$ can be improved exponentially, to $n^{1/\dim{X}}$.

Having realized that the statement in the previous edit was trivial (and uninteresting) as written, I just wanted to record those additional remarks here.

Answer by Vesselin Dimitrov for The height of an orbit under rational self-maps

2013-01-31T09:15:47Z

Thank you very much for this answer!

This is not going to fit in a comment, so I'm writing another answer.

[EDIT: I had hastily written some incorrect statements, now corrected in the text below. My apology for those mistakes. ]

I see from your conjecture that as soon as the dynamical degree $\delta_{\varphi} > 1$ and the orbit is Zariski dense, the height - conjecturally - ought to grow exponentially. On the other hand, when $\delta_{\varphi} = 1$, one needs a finer measure for the growth of both $\deg{\varphi^{n}}$ and $h(\varphi^n(x_0))$. Regarding the degree, we expect that as soon as $\deg{\varphi^n}$ is unbounded (which I guess will be the case as soon as $\varphi$ is not an birational?), it should grow at least linearly in $n$. But can one even show, unconditionally, that there exists an explicit function $\lambda(n)$, going monotonously to $\infty$, such that, for every $(X,\varphi)$ with $\deg{\varphi^n}$ unbounded, it holds $\deg{\varphi^n} > \lambda(n)$ for infinitely many $n$? I realize I can show this, say (for concreteness) with $\lambda(n) = \log{n}$.
Thus, my question mostly concerned the case $\delta_{\varphi} = 1$. It is of course possible, in that case, to have a Zariski-dense orbit with height growing logarithmically. (But I can prove in such a case, say for $X = \mathbb{P}^r$, if $h(\varphi^n(x_0)) = O(\log{n})$, then upon restricting $n$ to an arithmetic progression, the coordinates of $\varphi^n(x_0)$ are polynomials in $n$. In general, $\varphi$ must be birational. So this case is very special.)
Regarding the edit. I can actually show this with the $o(\log{n})$ term replaced by $o(n^{1/\mathrm{dim}(X)})$ --- but then the conclusion must be modified to include examples like translations of the affine line or automorphisms of $\mathbb{A}^2$ such as $(x,y) \mapsto (x+1,y+x^2)$. In effect: if $\varphi : X \dashrightarrow X$ is a rational self-map such that the orbit of $x_0$ is Zariski dense and contained in the domain of $\varphi$, and if $h(\varphi^n(x_0)) = o(n^{1/\dim(X)})$, then $\varphi$ is birational. (I wrote $o(\log{n})$ in my question only to retain the stronger conclusion that forces $x_0$ to be pre-periodic). To be sure, the same conclusion should be expected under the weaker bound $h(\varphi^n(x_0)) = o(n)$ --- but I can't prove this.
In particular: $\lambda(n) = \log{n}$ is the optimal function such that, for an arbitrary $(X,\varphi)$, the pre-periodic points can be characterized by $\lambda(n) - h(\varphi^n(x_0)) \to +\infty$.

Trichotomies in mathematics

2013-02-02T20:25:23Z

Added. Thanks to all who participated! Let me humbly apologize to those who were annoyed (quite understandably) by this thread, deeming it nothing more than an exercise in futility. If you thought the question, if legitimate at all, should have been restricted to interesting manifestations of a hyperbolic-parabolic-elliptic subdivision, then I can fully agree (although part of the idea was to interpret the question as you see fit); I left it open ended primarily because of the Weil trichotomy, which is of completely different kind and is so much more than a hierarchy, and in relation to which I was interested in hearing other people's opinions and elaborations. See, for instance, how Edward Frenkel, in a fascinating Bourbaki talk, builds upon the Weil trichotomy to introduce a parallel between Langlands and electro-magnetic dualities, which he uses as a springboard for the ideas from physics that have entered the arena of the geometric Langlands program. Or take the cherished three-sided parallel between the basic three-dimensional (from the point of view of etale cohomology) objects and their branched coverings: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathrm{Spec}(\mathbb{Z})$, and $S^3$, with primes in number fields corresponding to knots in threefolds, $\log{p}$ corresponding to hyperbolic length, etcetera.

To those who were not convinced that there was a neat trichotomy of algebraic surfaces (arguing they should instead form a tetrachotomy by Kodaira dimension), let alone in higher dimensional algebraic geometry, I refer to Sándor Kovács's answer here, which demonstrates rather eloquently the fundamental trichotomy of birational geometry:

http://mathoverflow.net/questions/81913/how-frequent-are-smooth-projective-varieties-with-anti-ample-canonical-bundle

Original post. For many purposes, notably in classification hierarchies or in Weil's "big picture" of the fundamental unity in mathematics, it seems as if mathematical reality is more accurately captured by trichotomies than by two-sided dictionaries or questions of "either/or." The most basic is of course the trichotomy negative - zero - positve embodied by the complete ordered field $(\mathbb{R},<)$ --- this is the Arrow of Time, if you will, or the conditioning of a dynamical system into states of past/present/future. As evidenced by some of the examples below, this trichotomy underlies varied, if crude, classification schemes in mathematics.

Other trichotomies arise from closer examinations of a mathematical parallel. Mathematicians have always been fond of discovery by analogy; they take very seriously the intuitions supplied by different yet loosely connected fields. In doing so, they are guided by a tacit, platonic belief in the fundamental Unity of mathematics. An example is the similarity between finite geometries and Riemann surfaces. To explain this parallel, indeed to make sense of it, it is necessary to provide a "middle column" in the dictionary: the arithmetic geometry of number fields and arithmetic surfaces. This leads to the trichotomy that Weil explained so lucidly in a letter (which he wrote in 1940 in prison for his refusal to serve in the army) to his sister, the philosopher Simone Weil. This point of view led, as we know, to an entire new field of mathematical inquiry.

Below I have listed some other cherished mathematical trichotomies. I am interested in seeing yet others, perhaps more specialized. This is my question: add more trichotomies to the list. Furthermore, I am interested in any reflections anyone might have, such as pertaining, for instance, to any of the following questions. Is 3 the most ubiquitous number in coarse classification schemes? Is it fair to say that a given trichotomy echoes the primeval trichotomy $(-,0,+)$? In a given trichotomy, is there a natural "middle column" of a corresponding three-sided dictionary? Is this "middle column" in any way the most fundamental, the most interesting, or the most elusive?

Trichotomies in mathematics: some examples.

The fabric of topology, geometry, and analysis is the real line $\mathbb{R}$. Tarski's eight axioms characterize it in terms of a complete binary total order <, a binary operation +, and a constant 1. (Multiplication comes afterwards - it is implied by Tarski's axioms - and so does the Bourbaki definition of the reals as the complete ordered field). The sign trichotomies $(<,=,>)$ and $(-,0,+)$ ensuing from those axioms have repercussions throughout all of mathematics.
For example, there are three constant curvature spaces, leading to the three maximally symmetric geometries: hyperbolic, flat (or Euclidean), and elliptic (e.g. spherical forms).
Locally symmetric spaces fall into three types: non-compact type, flat, and compact type.
In complex analysis, there are three simply connected cloths: the Riemann surfaces $\Delta$, $\mathbb{C}$, and $\hat{\mathbb{C}}$.
The connected component of the group of conformal automorphisms of a compact Riemann surface is one of the following three: trivial, $S^1 \times S^1$, $\mathrm{PGL}_2(\mathbb{C})$.
The complexity of fundamental groups, as showcased first of all by topological surfaces: genuinely non-abelian (perhaps we could say: anabelian) - abelian (or more generally, containing a finite index nilpotent subgroup) - and trivial (or more generally, finite). This is of course related to the subject of growth of finitely generated groups, brought forward by Lee Mosher's answer.
In dynamics, a fixed point (or a periodic cycle) can be either repelling, indifferent, or attracting.
In Thurston's work on surface homeomorphisms, elements of the mapping class group are classified according to dynamics into three types: pseudo-Anosov, reducible, and finite-order.
In algebraic geometry, the positivity of the canonical bundle is central to the classification and minimal model problems. More generally, positivity is a salient feature of algebraic geometry. For a delightful discussion, see Kollar's review of Lazarsfeld's book "Positivity in algebraic geometry." (Bull. AMS, vol. 43, no. 2, pp. 279-284). The most basic example is the trichotomy of algebraic curves (rational, elliptic, general type).
In birational algebraic geometry, at a very coarse level, there are three kinds of varieties out of which a general variety is made: rational curves, Calabi-Yau manifolds, and varieties of general type (or hyperbolic type, if you prefer). For example, an algebraic surface either: 1) admits a pencil of rational curves; or 2) admits a pencil of elliptic curves or is abelian or K3 (or a double quotient of a K3); or else 3) it is of general type. Abelian and K3 are examples of Calabi-Yau manifolds.
More concretely, consider smooth hypersurfaces $X \subset \mathbb{P}^n$. They divide into three types, according to how their degree $d$ compares with the dimension. If $d \leq n$, they contain plenty of rational curves (certainly uncountably many). If $d = n+1$, they are an example of a Calabi-Yau manifold, and typically contain a countably infinite number of rational curves. (The generating function of the number of rational curves of a given degree is then a very interesting function, of significance in the physics of quantum gravity.) And if $d \geq n+2$, then $X$ is of general type, and it is conjectured to typically contain only finitely many rational curves. (More precisely, Bombieri and Lang have conjectured that a variety of general type contains only finitely many maximal subvarieties not of general type).
In diophantine geometry, rational points are supposed to come from rational curves and abelian varieties. The sporadic examples are believed to be finitely many. This leads to the following trichotomy for the growth rate of the number of rational points of bounded (big, i.e. exponential) height: polynomial growth - logarithmic growth - $O(1)$. Furthermore, even in dimension 1, it is for abelian varieties that the situation is the deepest and the most mysterious.
In topology, it seems as if the interesting dimensions fall into three qualitatively different ranges: $d = 3$, $d = 4$, and $d \geq 5$. (Although this might be stretching it a bit too much). Of these, four dimensions -- the "middle column" -- is the most mysterious, and also the most relevant for physics.
The "Weil trichotomy," of course, goes at least as far back to Kronecker and Dedekind: curves over $\mathbb{F}_q$ - number fields - Riemann surfaces. Class field theory and Iwasawa theory are particularly eloquent examples of this trichotomy. Another example is of course the zeta function and the Riemann hypothesis.
One would be tempted to extend the latter trichotomy to [non-Archimedean world ($p$-adic, profinite) - global arithmetic - Archimedean world (geometry, topology, complex variables)], if the middle column did not subsume (much of) the flanking columns. Likewise the triple [$l$-adic cohomology-motive-Hodge structure] would probably not be admissible. Here is a variation on the theme (you may find it to be rubbish, in which case throw it away). There are two ways of completing (or taking limits of) the regular polygons $C_n$. The first is to think of $C_n$ as $\frac{1}{n}\mathbb{Z}/\mathbb{Z}$ and take the direct limit (in this case, union, or synthesis: $\rightarrow$), which is $\mathbb{Q}/\mathbb{Z}$. Completing, we get the circle $S^1 = \mathbb{R}/\mathbb{Z}$, which is the simplest manifold. The second is to think of $C_n$ as $\mathbb{Z}/n$ and take the projective limit (or deconstruction: $\leftarrow$), which is $\hat{\mathbb{Z}} = \prod_p \mathbb{Z}_p$, the profinite version of the circle. In this way, Archimedean (continuous) objects and $p$-adic objects may be seen as the two possible different limits (synthesis and deconstruction) of the same finite objects. Taking $C_n$ to be more general finite groups, we get essentially all the Lie groups, on the one hand; and all the profinite groups, on the other hand.
That we live in three perceptible spatial dimensions does not, of course, fit our bill. But in 1984, Manin published an article ("New dimensions in geometry") in which, guided by ideas from number theory (Arakelov geometry) and physics (supersymmetry), he proposed that there are three kinds of geometric dimensions, modeled on the affine superscheme $\mathrm{Spec} \mathbb{Z}[x_i;\xi_j]$, an "object of the category of topological spaces locally ringed by a sheaf of $\mathbb{Z}/2$-graded supercommutative rings." Here, $\xi_j$ are "odd," anticommuting variables, commuting with the "even" variables $x_i$. See the three coordinate axes $x, \xi$ and $\mathrm{Spec} \mathbb{Z}$ in his picture of "three-space-2000." The arithmetic axis $\mathrm{Spec} \mathbb{Z}$ is implicit in complex algebraic geometry, and is essential in problems such as the Ax-Grothendieck theorem and the construction of rational curves in Fano manifolds.
In the theory of linear groups there is, loosely speaking, a trichotomy: $\mathbb{G}_m$ (linear tori) - semisimple - $\mathbb{G}_a$ (unipotent).
Algebraic groups: reductive - abelian variety - unipotent. Especially, the classification of one-dimensional groups: $\mathbb{G}_m$ - $E$ - $\mathbb{G}_a$. (Thanks, Terry Tao!)
Variant: among commutative algebraic groups, there are: multiplicative type - abelian varieties - additive type (unipotent).
$\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ are the only finite-dimensional associative division algebras over the continuum. (Thanks Paul Reynolds, Teo B, and Sam Lewallen!)
The most basic PDEs of physics: the wave equation (hyperbolic) - the heat and Schrodinger equations (parabolic) - the Laplace equation (elliptic). (Thanks Alexandre Eremenko!)
An infinite finitely-generated group has $1,2$ or $\infty$ ends. (Thanks shane.orourke and Artie Prendergast-Smith!)
A random walk is either transient, null recurrent, or positive recurrent. (Thanks Vaughn Climenhaga!)
Zeta functions can by dynamical (Artin-Mazur); arithmetical on schemes of finite type over $\mathbb{Z}$ (Riemann and Hasse-Weil); and geometric (Selberg's zeta function of a hyperbolic surface).
In Model theory, there is an important trichotomy between super-stable theories, strict-stable (stable but not superstable) theories, and non stable theories.
It seems fair to say that there are three kinds of three-dimensional simply connected spaces: $\mathbb{P}_{\mathbb{F}_q}^1$, $\mathbb{Spec}(\mathbb{Z})$ compactified at archimedean infinity, and $S^3$. This brings about the Mazur knotty dictionary and the fruitful analogy between primes and knots (especially hyperbolic knots).

Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?

2013-02-09T11:43:13Z

Let $X$ be an $n$-dimensional compact Kahler manifold with negative first Chern class. Are its Chern numbers $\prod_{i=1}^{n-1} c_i^{k_i}$, over $k _i \geq 0$ with $\sum ik_i = n$, bounded in terms of $(-1)^n e(X) = (-1)^nc_n = c_n(\Omega_X^1)$?

Are there, moreover, only finitely many deformation types for $|e(X)| = (-1)^{n}e(X)$ bounded?

For $n = 2$, we have of course the famous Bogomolov-Miyaoka-Yau bound $c_1^2 \leq 3c_2$.

Remark. One may also include the convention $c_0 := 1$, for which the answer is positive since $(-1)^nc_n > 0$, in order to cover the prototypical observation that the Euler number of a hyperbolic surface is negative. In view of this example, I would also like to extend the original question to the realm of log manifolds and hyperbolicity.

How many curves in a family possess a rational point?

2013-02-07T18:22:07Z

Consider $B$ a finite-type integral quasi-projective scheme over $\mathbb{Z}$ such that $B(\mathbb{Z})$ infinite. (If you like, take $B$ to be the affine line). Let $X \to B$ be a generically smooth proper family of curves of genus $g > 1$. Assume the $b \in B(\mathbb{Z})$ for which $X_b(\mathbb{Q}) = \emptyset$ are Zariski-dense in $B$. Must then the proportion of members $X_b$, over $b \in B(\mathbb{Z})$, for which $X_b(\mathbb{Q}) \neq \emptyset$, be equal to $0$? (Variant: the same question with $B(\mathbb{Z})$ replaced by $B(\mathbb{Q})$.)

An explicit variant (although not quite a special case as it stands). Let $f \in \mathbb{Z}[x]$ be irreducible of degree $> 4$. Do the integers $N$ for which $f(x) = Ny^2$ has a rational solution, have density zero? Variant: let $N$ range over the primes, or over the squarefrees.

In the above setup, we may include this example by assuming more generally that $X_{\mathbb{Q}} \to B_{\mathbb{Q}}$ admits exactly $m$ sections defined over $\mathbb{Q}$, and ask whether the density of $b \in B(\mathbb{Z})$ with $|X_b(\mathbb{Q})| > m$ is zero. Alternatively, drop the properness and generic smoothness assumptions, including instead the assumption that the smooth projective model of the generic fibre has genus $> 1$.

Answer by Vesselin Dimitrov for Counting higher dimensional abelian varieties of a given conductor

2013-02-09T11:25:02Z

Here is a variant of this problem. Consider the number of $g$-dimensional p.p. abelian varieties with everywhere good reduction over a varying number field $K$. In genus $g = 1$, it follows from the uniform finiteness theorem for the unit equation that this number is bounded solely in terms of $[K:\mathbb{Q}]$. If you consider more generally elliptic curves on $K$ with good reduction outside $S$, the bound becomes simply exponential in $[K:\mathbb{Q}] + |S|$.

Are there any results available, for higher genus $g > 1$, on whether or not the number of $g$-dimensional p.p. abelian schemes on $R := \mathcal{O}_{K,S}$ could be bounded solely in terms of $g$ and the rank of $R$?

Average ranks of abelian surfaces

2013-02-05T23:38:38Z

Most people nowadays believe that over a fixed global field, $50$% of the elliptic curves have $0$ rank, $50$% have rank $1$, and $0$% have higher rank. A significant advance in this direction has been Bhargava and Shankar's proof that for elliptic curves over $\mathbb{Q}$, the rank is bounded on average (indeed, not just the Mordell-Weil rank, but even the $2$-Selmer rank).

Has anybody put forth a similar guess for the ranks of higher dimensional abelian varieties? If so, what is the rationale behind such a guess? I am asking about the precise statistical distribution of ranks.

Answer by Vesselin Dimitrov for Modular Forms - Eichler quote

2013-02-03T16:35:22Z

My interpretation of the quote is different. (Granted, whether Eichler really said such a thing is a different question). It should refer to solving the algebraic equations, to the legacy of Abel and Galois and to Kronecker's Jugendtraum! To the book of Taniyama and Shimura, whereby certain abelian equations are solved by adjoining the moduli of a relevant abelian variety. In this sense, from the point of view of exact solutions to algebraic equations, the modular functions such as $j$ are indeed a natural, inevitable complement to the basic arithmetic operations $+, \times, \sqrt[n]{\cdot}$. This falls into the rubric of explicit class field theory. For a connection to modular forms such as $\Delta(\tau)$, see Ribet's converse to Herbrandt's theorem; a great introduction to this circle of ideas is Mazur's article, "How can we construct abelian Galois extensions of basic number fields?" (Bull. AMS, vol. 48, no. 2, pp. 155-209).

Answer by Vesselin Dimitrov for Are rational varieties simply connected?

2013-01-31T18:56:48Z

Yes! (I assume it was implicit in your question that the variety be projective?)

More generally: any smooth, complex, rationally connected projective variety is simply connected. See Debarre's book ("Higher dimensional algebraic geometry").

Alternatively, take a look at Debarre's Bourbaki talk ("Varietes rationnellement connexes"). The idea is that a rationally connected variety has no holomorphic forms, so by Hodge theory the structure sheaf $\mathcal{O}_X$ is acyclic, implying $\chi(X,\mathcal{O}_X) = 1$. If $f : Y \to X$ is a connected etale cover, then $Y$ is again rationally connected, so by the same argument $\chi(Y,\mathcal{O}_Y) = 1$, so $\deg{f} = 1$. So $X$ is simply connected.

In positive characteristic the Hodge theory fails, so the argument as such doesn't stand, but you may still get the simple connectedness as a consequence of the fibration theorem of Graber-Harris-Starr-de Jong. See the mentioned Bourbaki expose.

Answer by Vesselin Dimitrov for Philosophy behind Mochizuki's work on the ABC conjecture

2012-09-15T18:49:05Z

Last revision: 10/20. (Probably the last for at least some time to come: until Mochizuki uploads his revisions of IUTT-III and IUTT-IV. My apology for the multiple revisions. )

Completely rewritten. (9/26)

It seems indeed that nothing like Theorem 1.10 from Mochizuki's IUTT-IV could hold.

Here is an infinite set of counterexamples, assuming for convenience two standard conjectures (the first being in fact a consequence of ABC), that contradict Thm. 1.10 very badly.

Assumptions:

A (Consequence of ABC) For all but finitely many elliptic curves over $\mathbb{Q}$, the conductor $N$ and the minimal discriminant $\Delta$ satisfy $\log{|\Delta|} < (\log{N})^2$.
B (Uniform Serre Open Image conjecture) For each $d \in \mathbb{N}$, there is a constant $c(d) < \infty$ such that for every number field $F/\mathbb{Q}$ with $[F:\mathbb{Q}] \leq d$, and every non-CM elliptic curve $E$ over $F$, and every prime $\ell \geq c(d)$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2(\mathbb{Z}/{\ell})$. (In fact, it is sufficient to take the weaker version in which $F$ is held fixed. )

Further, as far as I can tell from the proof of Theorem 1.10 of IUTTIV, the only reason for taking $F := F_{\mathrm{tpd}}\big( \sqrt{-1}, E_{F_{\mathrm{tpd}}}[3\cdot 5] \big)$ --- rather than simply $F := F_{\mathrm{tpd}}(\sqrt{-1})$ --- was to ensure that $E$ has semistable reduction over $F$. Since I will only work in what follows with semistable elliptic curves over $\mathbb{Q}$, I will assume, for a mild technical convenience in the examples below, that for elliptic curves already semistable over $F_{\mathrm{tpd}}$, we may actually take $F := F_{\mathrm{tpd}}(\sqrt{-1})$ in Theorem 1.10.

The infinite set of counterexamples. They come from Masser's paper [Masser: Note on a conjecture of Szpiro, Asterisque 1990], as follows. Masser has produced an infinite set of Frey-Hellougarch (i.e., semistable and with rational 2-torsion) elliptic curves over $\mathbb{Q}$ whose conductor $N$ and minimal discriminant $\Delta$ satisfy $$ (1) \hspace{3cm} \frac{1}{6}\log{|\Delta|} \geq \log{N} + \frac{\sqrt{\log{N}}}{\log{\log{N}}}. $$ (Thus, $N$ in these examples may be taken arbitrarily large. ) By (A) above, taking $N$ big enough will ensure that $$ (2) \hspace{3cm} \log{|\Delta|} < (\log{N})^2. $$ Next, the sum of the logarithms of the primes in the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ is $2(\log{N})^2 + o((\log{N})^2)$, so it is certainly $> (\log{N})^2$ for $N \gg 0$ big enough. Thus, by (2), it is easy to see that the interval $\big( (\log{N})^2, 3(\log{N})^2 \big)$ contains a prime $\ell$ which divides neither $|\Delta|$ nor any of the exponents $\alpha = \mathrm{ord}_p(\Delta)$ in the prime factorization $|\Delta| = \prod p^{\alpha}$ of $|\Delta|$.

Consider now the pair $(E,\ell)$: it has $F_{\mathrm{mod}} = \mathbb{Q}$, and since $E$ has rational $2$-torsion, $F_{\mathrm{tpd}} = \mathbb{Q}$ as well. Let $F := \mathbb{Q} \big( \sqrt{-1}\big)$. I claim that, upon taking $N$ big enough, the pair $(E_F,\ell)$ arises from an initial $\Theta$-datum as in IUTT-I, Definition 3.1. Indeed:

Certainly (a), (e), (f) of IUTT-I, Def. 3.1 are satisfied (with appropriate $\underline{\mathbb{V}}, \, \underline{\epsilon}$);
(b) of IUTT-I, Def. 3.1 is satisfied since by construction $E$ is semistable over $\mathbb{Q}$;
(c) of IUTT-I, Def. 3.1 is satisfied, in view of (B) above and the choice of $\ell$, as soon as $N \gg 0$ is big enough (recall that $\ell > (\log{N})^2$ by construction!), and by the observation that, for $v$ a place of $F = \mathbb{Q}(\sqrt{-1})$, the order of the $v$-adic $q$-parameter of $E$ equals $\mathrm{ord}_v (\Delta)$, which equals $\mathrm{ord}_p(\Delta)$ for $v \mid p > 2$, and $2\cdot\mathrm{ord}_2(\Delta)$ for $v \mid 2$;

while $\mathbb{V}_{\mathrm{mod}}^{\mathrm{bad}}$ consists of the primes dividing $\Delta$;

Finally, (d) of IUTT-I, Def. 3.1 is satisfied upon excluding at most four of Masser's examples $E$. (See page 37 of IUTT-IV).

Now, take $\epsilon := \big( \log{N} \big)^{-2}$ in Theorem 1.10 of IUTT-IV; this is certainly permissible for $N \gg 0$ large enough. I claim that the conclusion of Theorem 1.10 contradicts (1) as soon as $N \gg 0$ is large enough.

For note that Mochizuki's quantity $\log(\mathfrak{q})$ is precisely $\log{|\Delta|}$ (reference: see e.g. Szpiro's article in the Grothendieck Festschrift, vol. 3); his $\log{(\mathfrak{d}^{\mathrm{tpd}})}$ is zero; his $d_{\mathrm{mod}}$ is $1$; and his $\log{(\mathfrak{f}^{\mathrm{tpd}})}$ is our $\log{N}$. By construction, our choice $\epsilon := \big( \log{N} \big)^{-2}$ then makes $1/\ell < \epsilon$ and $\ell < 3/\epsilon$, whence the finaly display of Theorem 1.10 would yield $$ \frac{1}{6} \log{|\Delta|} \leq (1+29\epsilon) \cdot \log{N} + 2\log{(3\epsilon^{-8})} < \log{N} + 16\log{\log{N}} + 32, $$ where we have used $\epsilon \log{N} = (\log{N})^{-1} < 1$ for $N > 3$, and $2\log{3} < 3$.

The last display contradicts (1) as soon as $N \gg 0$ is big enough.

Thus Masser's examples yield infinitely many counterexamples to Theorem 1.10 of IUTT-IV (as presently written).

Added on 10/15, and revised 10/20. Mochizuki has commented on the apparent contradiction between Masser's examples and Theorem 1.10:

http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf

He writes that he will revise portions of IUTT-III and IUTT-IV, and will make them available in the near future. (He estimates January 2013 to be a reasonable period). He confirms the following ["essentially"] anticipated revision of Theorem 1.10:

Let $E/\mathbb{Q}$ be a semistable elliptic curve with [say, for the sake of simplifying] rational $2$-torsion [i.e., a Frey-Hellegouarch curve] of minimal discriminant $\Delta$ and conductor $N$ (square-free). For $\epsilon > 0$, let $N_{\epsilon} := \prod_{p \mid N, p < \epsilon^{-1}} p$. Then: $$ \frac{1}{6} \log{|\Delta|} < \big( 1 + \epsilon \big) \log{N} + \Big( \omega(N_{\epsilon}) \cdot \log{(1/\epsilon)} - \log{N_{\epsilon}} \Big) + O\big( \log{(1/\epsilon)} \big) $$ $$ < \log{N} + \Big( \epsilon \log{N} + \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big) + o\Big( \big( \epsilon \log{(1/\epsilon)} \big)^{-1} \Big), $$ where $\omega(\cdot)$ denotes "number of prime factors." The second estimate comes from the prime number theorem in the form $\pi(t) = t/\log{t} + t/(\log{t})^2 + o\big( t/(\log{t})^2 \big)$, applied to $t := \epsilon^{-1}$, and is sharp if you restrict $\epsilon$ to the range $\epsilon^{-1} < (\log{N})^{\xi}$ with $\xi < 1$, as there nothing prevents $N$ from being divisible by all primes $p < (\log{N})^{\xi}$. In particular, as the Erdos-Stewart-Tijdeman-Masser construction is based on the pigeonhole principle, which cannot preclude that $N$ be divisible by all the primes $< (\log{N})^{2/3}$, the second estimate could very well be sharp in all the Masser examples. As it is easily seen that the bracketed term exceeds the range $\sqrt{\log{N}}/(\log{\log{N}})$ of Masser's examples, this has the implication that

the Erdos-Stewart-Tijdeman-Masser method cannot disprove Mochizuki's revised inequality,

which therefore seems reasonable.

On the other hand, if we take $\epsilon := (\log{N})^{-1}$ and assume $\omega(N_{\epsilon})$ bounded, this would yield $(1/6)\log{|\Delta|} < \log{N} + O(\log{\log{N}})$, just as before. (Thus, Mochizuki predicts that this last bound must hold for $N$ a large enough square-free integer such that the number of primes $< \log{N}$ dividing $N$ is bounded. I cannot see evidence neither for nor against this at the moment: again, the Masser and Erdos-Stewart-Tijdeman constructions are based on the pigeonhole principle, and do not seem to be able to exclude the small primes $< \log{N}$. So here we have an open problem by which one could probe Mochizuki's revised inequality. A reminder: in terms of the $abc$-triple, $\Delta$ is essentially $(abc)^2$, and $N = \mathrm{rad}(abc)$).

A side remark: note that the inverse $1/\ell$ of the prime level from the de Rham-Etale correspondence $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$ in Mochizuki's "Hodge-Arakelov theory" ultimately figures as the $\epsilon$ in the ABC conjecture.

[I have deleted the remainder of the 10/15 Addendum, since it is now obsolete after Mochizuki's revised comments. ]

Answer by Vesselin Dimitrov for Topics for an Undergraduate Expository Paper in Number Theory

2012-09-25T22:32:55Z

The transcendence of $2^{\sqrt{2}}$ and $e^{\pi}$: Gel'fond's proof. (Assuming some basic complex analysis).
Nathanson's problem: show that $3^n \nmid 5^n-2$ for $n > 1$. (This involves the $p$-adic analog of the above topic).
More elementary (see also Yuhao Huang's answer above): the determination of $F_p \mod{p}$ for the Fibonacci sequence (i.e., periodic modulo $5$), as a consequence of the congruence $2\cos{(px)} \equiv 2\cos^p(x) \mod{p\mathbb{Z}[\cos{x}]}$ and the formula $(1+\sqrt{5})/4 = \cos{(2\pi/10)}$. This I think is a good, algebraic point of entry into quadratic reciprocity (obviously, it is equivalent to the splitting law for $\mathbb{Q}(\sqrt{5})$). A key point of course is to explain that $1/p$ is not a sum of roots of 1 (or, if one prefers, that is not an albgebraic integer), so that the congruence may be exploited appropriately. Interpretation as a "Fermat's little theorem" for $\mathbb{Q}(\sqrt{5})$.
Related: Exhibit a formula showing that $\sqrt{N} \in \mathbb{Q}\big( e^{2\pi i/4N} \big)$, and perhaps use this to conclude that the residue of $N^{\frac{p-1}{2}} \mod{p}$ only depends on the residue of $p \mod{4N}$.
$\mathbb{Q}$ has no unramified extensions.
There is always a prime between $n$ and $2n$: Erdos' elementary proof.
The Wolstenholme-Jacobsthal congruence $\binom{np}{mp} \equiv \binom{n}{m} \mod{p^3}$ using the "Stirling formula" for the $p$-adic $\Gamma$-function. Or combinatorial proofs of such congruences. (Or indeed, any other congruence from A. Granville's "Arithmetic Properties of Binomial Coefficients.")
Completely elementary: Zsygmondy's theorem and applications. (Here is one: find all integer solutions of $a^n = b^n + c^k$ subject to $|c| \leq n$).
If $a^n - 1 \mid b^n - 1$ for all $n > 0$, then $b = a^j$. If $a4^n + b6^n + c9^n$ is a perfect square for each $n$, then $(a,b,c) = (r^2,2rs,s^2)$. Solve, and generalize both!

Answer by Vesselin Dimitrov for Are certain simple Lie groups linear algebraic groups?

2012-09-19T15:56:13Z

The answer is yes for complex Lie groups, and follows from the classification. (Root data are in fact defined over $\mathbb{Z}$: a complex semisimple group has not only an underlying algebraic, but even an arithmetic structure). For a more direct explanation, see Theorem 6.3 in the book "Lie Groups and Lie Algebras III" by Onischik-Vinberg: any connected complex Lie group satisfying $G = [G,G]$ and admitting a faithful linear representation (which for semisimple groups is automatic), has a unique underlying complex algebraic structure.

For real Lie groups this is not quite true, as noted above.

Answer by Vesselin Dimitrov for Philosophy behind Mochizuki's work on the ABC conjecture

2012-09-17T14:34:42Z

Let me also try to give, in a modest complement to Minhyong Kim's great post, some additional remarks on Mochizuki's strategy. The idea that has led to the development of "Inter-universal Teichmuller theory for number fields" is certainly very beautiful, and was known to Mochizuki, along with the nature of the final estimate, already in 2000. (But let us recall, as a sane reminder of just how elusive the ABC conjecture has been, Miyaoka's flawed proof: did the idea of a Bogomolov-Miyaoka-Yau type bound involving arithmetic Chern numbers in Arakelov theory not seem equally beautiful, exciting, and promising?)

In brief, the main idea behind the IUTT-series is to construct, outside the rigid confines of algebraic geometry, a subtle object simulating a rank-1, Galois-stable quotient of $E[\ell]$. Here, $E/\mathbb{Q}$ is a (pretty much) arbitrary rational elliptic curve (and this is the main point: such a Galois-stable quotient will almost never exist!); and $\ell \geq 5$ is an auxiliary prime, generic for $E$ in a very mild sense, but otherwise free to optimize until the very final estimate. This is then applied to construct, in the non-linear discretized "Hodge-Arakelov theory," a comparison isomorphism between $(E^{\dagger}, <\ell)$ and $E[\ell]$, which is free of Gaussian poles at the bad places of $E$. For this then leads to a promising Galois-theoretic "Kodaira-Spencer map," as explained in Minhyong Kim's post, hopefully leading in the familiar way to the arithmetic Szpiro inequality for this very same elliptic curve: $\log{|\Delta_{\mathrm{min}}(E)|} \leq (6+\varepsilon) \log{N_E} + O_{\varepsilon}(1)$.

Let me, however, disagree with one point from M. Kim's post. My impression is that what Mochizuki calls an "initial $\Theta$-datum" - and which is, essentially, the pair of the rational elliptic curve $E$ (or equivalently, the $abc$-triple from the ABC-conjecture!) and, until the very final estimate in Ch. 2 of the fourth paper, the prime level $\ell$ - are fixed for good throughout the entire series of IUTT-papers. The deformation flavor of "Teichmuller theory" refers to dismantling the underlying number field, and not to the elliptic curve enhancement (indeed, in Mochizuki's dictionary with his own $p$-adic Teichmuller theory, it is the number field that corresponds to a hyperbolic curve; the elliptic curve enhancement corresponds to an "indigenous bundle" over the hyperbolic curve, and invites the anabelian philosophy via the \'etale fundamental group of the once-punctured elliptic curve). All the "Hodge theaters" associated to the initial $\Theta$-datum are isomorphic to one another, and form a vastly complicated $2$-dimensional non-commutative array - the "$\mathfrak{log}-\Theta$ lattice" - of non-ring theoretic translations between one another. What Mochizuki writes on p. 10 of IUTT-I is that the theory of $\Theta$-Hodge theaters "may be regarded as a sort of solution to the problem of constructing the global quotient $E[\ell] \twoheadrightarrow Q$" [needed for the application to arithmetic Kodaira-Spencer]. He does not seem to suggest that this is done by "moving the initial $E$ to a single elliptic curve via the intermediate case of an elliptic curve in general position," as M. Kim writes. (The term "elliptic curves in general position" indeed figures in Mochizuki's fourth paper, but it has a different, not-so-essential significance that comes through his entirely self-contained paper [GenEll], and whose purely technical purpose is to reduce the general ABC conjecture to the restricted version of Szpiro's inequality for $E$, in Thm. 1.10 of IUTT-IV, coming from the estimate in IUTT-III).

In particular, in sharp contrast to the Thue-Siegel-Roth tradition of Diophantine approximations, Mochizuki's program does not seem to compare different elliptic curves / $abc$-triples, all the way through to the key estimate $$ (*) \hspace{3cm} \log{|\Delta_{\mathrm{min}}(E)|} \leq \big(6 + \varepsilon + 200/\ell\big)\log{N_E} + 12\log(\ell\varepsilon^{-7}) $$ of IUTT-IV [asserted for all primes $\ell \geq 5$ that are generic for $E$ in a rather mild sense: essentially, $\ell$ has to be prime to the degenerate places and the $q$-parameters of $E$. Also, $\varepsilon \in (0,\epsilon_0)$ is arbitrary, with $\epsilon_0$ a numerical value. ] In this sense, Mochizuki's approach - nevermind the vast technical difficulties precipitated by the non-ring theoretic simulation of a global quotient $E[\ell] \twoheadrightarrow Q$ - is entirely direct and, consequently, effective.

So what does Mochizuki actually (claim to) prove?

Start with an $abc$-triple (co-prime rational integers with $a+b+c=0$). Since the discriminant $(\alpha-\beta)(\beta-\gamma)(\gamma-\alpha)$ of a cubic polynomial $x^3 + \cdots$ encapsulates exactly this equation, it is a profitable, traditional idea to interpret the $abc$-datum as the giving of the rational elliptic curve $E = E_{a,b,c}$ defined by the equation $y^2 = x(x-a)(x+b)$. The (apparently weaker, but virtually as powerful) ABC conjecture $abc < K_{\varepsilon}\cdot\mathrm{rad(abc)}^{3+\varepsilon}$ then translates into Szpiro's inequality: $\log{|\Delta_{\min}(E)|} \leq (6+\varepsilon)\log{N_E} + O_{\varepsilon}(1)$ between the minimal discriminant $\Delta_{\min}(E)$ and conductor $N_E$ of $E$ (which are, essentially, $(abc)^2$ and $\mathrm{rad}(abc)$). Pick the "auxiliary prime" $\ell \geq 5$ to be generic for $E$ in the sense that, essentially: (1) $\ell \nmid abc$; (2) $\ell$ does not divide the prime exponents in $abc$; (3) for $F := \mathbb{Q}( \sqrt{-1}, E[15] )$, the Galois representation of $G_F$ on $E[\ell]$ has full image $\mathrm{GL}_2 (\mathbb{Z}/\ell)$. [Conjecturally, the last condition should only exclude a finite list of primes, independent of $E$!] Then Mochizuki [IUTT-IV, Thm. 1.10] claims that (*) should hold for any $\varepsilon < \epsilon_0$.

This is the essential Diophantine estimate. Anything further than that [i.e., the deduction of the full ABC conjecture in IUTT-IV, Section 2] consists of standard, and relatively straightforward reductions [such as, e.g., the use of non-critical Belyi maps] elaborated in Mochizuki's self-contained paper [GenEll]: "Arithmetic elliptic curves in general position." Mochizuki indeed writes, in his first paper, that the auxiliary prime level $\ell \geq 5$ from the Hodge-Arakelov discretized non-linear comparison isomorphisms/correspondences $(E^{\dagger}, < \ell) \leftrightarrow E[\ell]$, will be chosen in the Diophantine application to be large, roughly on the order of the height of $E$. But this comes entirely through Theorem 3.8 in [GenEll]: there, the various non-divisibility properties are ensured by simply taking $\ell$ to exceed all the primes of bad reduction / all the $q$-parameters (also, the full Galois action is ensured unconditionally). In (*), $\ell$ could be any prime satisfying the mentioned non-divisibility conditions. (This, by the way, is what I considered highly disturbing).

My apology if I have misunderstood - and misrepresented - the points from Mochizuki's papers that I have alluded to.

Answer by Vesselin Dimitrov for How do I find the set of all lines lying on a general quadric in $\mathbb{CP}^3$?

2012-09-15T23:11:56Z

You can show that a smooth quadric in $\mathbb{P}^3$ is GL-equivalent to the product $\mathbb{P}^1 \times \mathbb{P}^1$ embedded by the Segre map. Once you know this, you're done. (A curve on $\mathbb{P}^1 \times \mathbb{P}^1$ has a bidegree $(a,b)$, and its degree in $\mathbb{P}^3$ under the embedding $\mathbb{P}^1 \times \mathbb{P}^1 \hookrightarrow \mathbb{P}^3$ is $a+b$. Thus the lines fall in two families, corresponding to the bidegrees $(1,0)$ and $(0,1)$. )

Comment by Vesselin Dimitrov

2013-03-07T18:55:38Z

Regarding the addendum: Thank you very much for this observation! Indeed it works for $r$ independent points; and this should yield, as in the paper by Amoroso and David, the truth of the original elliptic Lehmer conjecture for points $P$ such that $\mathbb{Q}(P)/\mathbb{Q}$ is Galois. I wonder if Masser (or someone else) has a similar counting theorem for points of small height on higher dimensional abelian varieties (thus refining his estimate $\hat{h}(P) > cd^{-\kappa}$)?

Comment by Vesselin Dimitrov

2013-03-06T00:01:49Z

@ACL: Thanks! I knew about that paper, but I had not looked at it, so I didn't know this question was formulated as a conjecture there. It seems as if there has been no progress on this problem for $r > 1$?

Comment by Vesselin Dimitrov

2013-03-05T21:14:32Z

Lang's conjecture demands a uniform $c$, independent of $E$. Here I ask for a $c$ depending on both $E$ and $r$.

Comment by Vesselin Dimitrov

2013-03-02T16:23:33Z

Thank you very much for this reference, that was very helpful! That is exactly what I wanted to know. By the way, the reference also indicates that $1/2$ is the critical exponent for which one can prove zero density unconditionally, i.e. without assuming GRH. (They refer to the paper by Pappalardi, where this was shown.)

Comment by Vesselin Dimitrov

2013-03-02T06:57:42Z

Thank you very much indeed for all those explanations, and especially for the references!

Comment by Vesselin Dimitrov

2013-02-28T11:18:50Z

Thank you very much for those explanations! Thus, the [SUZ] equidist. thm. implies that the lim inf (under Zariski topology) of the height of a totally real point is bounded away from zero; but to get a lower bound over all non-torsion totally real points, you need Ullmo's idea. And to get an effective lower bound on the height of a non-torsion point, you need the David-Philippon analytic geometry approach. However, the bound on the lim inf from [SUZ] is already effective - do I understand this right? Thus, I wondered whether your p-adic equid. thm. would likewise give effective $U$ and $c$.

Comment by Vesselin Dimitrov

2013-02-27T23:47:58Z

Thanks a lot, this was very helpful! I will look into those two papers. Is there any lower bound on the number $c$ that you get from your equidistribution theorem? E.g., when you fix $A$, how does the optimal $c$ vary with $p$? Is it always $> \epsilon/p$ as in the $\mathbb{G}_m$ case, with $\epsilon > 0$ a constant independent of $p$?

Comment by Vesselin Dimitrov

2013-02-27T16:06:56Z

By the way, Hindry and Silverman have shown that for elliptic curves with integral modulus (everywhere good reduction), the number of torsion points rational over a given number field of degree $d$ is at most $\mathrm{const} \cdot d\log{d}$. This lends, perhaps, additional support to the expectation that the bound in Merel's theorem should be polynomial. (Note that the trivial bound provided by good reduction at $2$ is exponential in $d$.)

Comment by Vesselin Dimitrov

2013-02-27T14:22:22Z

@Michael: This is stated, for example, in Remark 2 in Marusia Rebolledo's expository article ("Merel's theorem on the boundedness of the torsion of elliptic curves.") I am sure I have seen it in at least one other paper, but right now I can't remember the reference. But the idea is that since for individual elliptic curves the count is proportional to $[K:\mathbb{Q}]^{3/2}$ and $[K:\mathbb{Q}]^2$ in the non-CM and CM cases, respectively, it is not too wild to ask whether this count is uniform. Thus, one can ask whether the count is in fact uniformly bounded by $C(r) [K:\mathbb{Q}]^{2+r}$.

Comment by Vesselin Dimitrov

2013-02-27T13:57:13Z

As far as I know, the conclusion of Merel's theorem is believed to hold for abelian varieties up to bounded dimension. In fact, the bound on the $K$-rational torsion is supposed to be polynomial in $[K:\mathbb{Q}]$. (This is wide open even for elliptic curves; Merel's theorem gives an exponential bound.)

Comment by Vesselin Dimitrov

2013-02-12T16:23:53Z

Take a look at Gromov's short essay for Mariana Cook's photo album, "Mathematicians: An outer view of the inner world." You may (or may not...) find his reflection illuminating; and I think it is relevant to your question.

Comment by Vesselin Dimitrov

2013-02-12T10:35:11Z

Thank you very much, this paper is really helpful!

Comment by Vesselin Dimitrov

2013-02-09T23:34:57Z

Come to think of it, I even think it is indeed better to delete it. The discussion itself was very useful - thanks a lot! As for the result, we may write it up cleanly (e.g. I could include it inside the original question, if it ever comes up).

Comment by Vesselin Dimitrov

2013-02-09T23:30:56Z

You mean the contrapositive of strict harmonicity :-). You may of course delete the answer, if you like. It's up to you; with me it's perfectly fine.

Comment by Vesselin Dimitrov

2013-02-09T23:22:15Z

By the way, could you please, at your leisure, edit your answer, so as to prevent potential misunderstanding?