Schedule for: 22w5024 - Specialisation and Effectiveness in Number Theory

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

The talk aims to present recent progress in the area Hilbert's irreducibility theorem. The theorem may be formulated as the statement there are ``many" of rational points on the *line* not coming from rational points on a given cover. If one replaces the line by other variety, the situation becomes more complicated and, in particular, there are obstructions to the theorem coming from rational points of unramified covers (recall that the line has no nontrivial ramified covers). The first result, j/w Daniele Garzoni, deals with affine groups: We show that a random walk on a finitely generated Zariski dense subgroup almost surely misses rational points coming from a ramified cover. The second result, j/w Arno Fehm and Sebastian Petersen, deals with abelian varieties and with rational points over the maximal cyclotomic field (or more generally, the field obtained by adding the torsion points of an abelian variety).

Let $K$ be a number field and let $\mathcal{E}=\mathcal{E}_K$ be the ring of $K$-power sums (i.e. functions of the shape $n\mapsto a_1\alpha_1^n+\cdots+a_t\alpha_t^n$ with coefficients $a_1,\ldots,a_t$ and characteristic roots $\alpha_1,\ldots,\alpha_t$ belonging to $K$). Moreover, let $f(n,X)$ be a polynomial in $X$ with coefficients in $\mathcal{E}$. In this talk we discuss the question, what can be said if $f$ specializes to a reducible polynomial in $K[X]$ for infinitely many $n$. Under suitable, but restrictive, assumptions we show that this happens if and only if $f$ is reducible as a polynomial in $\mathcal{E}[X]$, which can be checked effectively. This is joint work with Sebastian Heintze (TU Graz).

Irreducibility of random polynomials of large degree has been studied recently in works by several authors (in particular by Bary-Soroker, Kozma, Koukoulopoulos and by Varju and myself). We study analogous problems in the setting of word maps in matrix groups, such as $\mathrm{SL}_2(\mathbb C)$ or more general semisimple Lie groups. Conditionally on GRH, we are able to determine the dimension and number of components of word varieties with an exponentially small probability of exceptions. The proofs use effective Chebotarev type theorems and spectral gap bounds for Cayley graphs of finite simple groups. Joint work with Peter Varju and Oren Becker.

We prove in particular that for any sufficiently large prime $p$ there is $1\le a$<$p$ such that all partial quotients of $a/p$ are bounded by $O(\log p/\log \log p)$. This improves the well--known Korobov bound concerning Zaremba's conjecture from the theory of continued fractions.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

The Markoff surface $X : x^2 + y^2 + z^2 - xyz = 0$ first appeared in the work of Markoff in 1879 as part of his study of Diophantine approximation and binary quadratic forms. Since then, questions about the integral solutions of $X$ have been related to questions in numerous other settings, including the lengths of geodesics on hyperbolic tori, the monodromy of Painleve VI differential equations, and the derived categories of algebraic varieties. In this talk we will describe recent progress on a question of "abundance" of its integral solutions. Specifically, we will discuss a conjecture of Baragar (1991) and Bourgain, Gamburd, and Sarnak (2016) that the reduction map $X(\mathbb{Z})\rightarrow X(\mathbb{F}_p)$ is surjective for every prime $p$ (in this case we say that $X$ satisfies "strong approximation"). In 2016, using analytic methods, Bourgain, Gamburd, and Sarnak were able to establish the conjecture for all but a thin (but possibly infinite) set of primes $p$. In this talk we will describe how to promote "all but a thin set" to "all but an explicit finite set", thus reducing the conjecture to a finite computation. The key ingredient is a new "rigidity" coming from algebraic geometry: we will relate $\mathbb{F}_p$-points of $X$ to $\text{SL}_2(\mathbb{F}_p)$-covers of elliptic curves branched over the origin. By studying the "Hurwitz" moduli stack of such covers, we will show that if the reduction map is not surjective, then the complement of its image must have size at least linear in $p$. Since asymptotic results of Bourgain, Gamburd, and Sarnak prohibit this latter possibility, we deduce surjectivity for large $p$. The connection with Hurwitz stacks uses the fact that $X$ is a character variety for $\text{SL}_2$-representations of a free group of rank 2, and the method hints at an interesting relationship between the Diophantine properties of character varieties and the geometry of Hurwitz stacks.

In 1988, Erdős asked in Banff: let $x$ and $y$ be positive integers such that for all $n$, the set of primes dividing $x^n-1$ is equal to the set of primes dividing $y^n-1$. Is $x = y$? Corrales-Rodrigáñez and Schoof answered this question in the affirmative and showed more generally that, if every prime dividing $x^n-1$ also divides $y^n-1$, then $y$ is a power of $x$. In joint work with Francesco Campagna, we have studied this so-called support problem with the Hilbert class polynomials $H_D(T)$ instead of the polynomials $T^n-1$, replacing roots of unity by singular moduli. In my talk, I will state the result we obtained, sketch its proof, and tell you about a surprising property of the two singular moduli of discriminant $-15$ that we discovered.

The celebrated Skolem-Mahler-Lech Theorem states that the set of zeros of a linear recurrence sequence is the union of a finite set and finitely many arithmetic progressions. The corresponding computational question, the Skolem Problem, asks to determine whether a given linear recurrence sequence has a zero term. Although the Skolem-Mahler-Lech Theorem is almost 90 years old, decidability of the Skolem Problem remains open. The main contribution of this talk is to present an algorithm to solve the Skolem Problem for simple linear recurrence sequences (those with simple characteristic roots). Whenever the algorithm terminates, it produces a stand-alone certificate that its output is correct -- a set of zeros together with a collection of witnesses that no further zeros exist. We give a proof that the algorithm always terminates assuming two classical number-theoretic conjectures: the Skolem Conjecture (also known as the Exponential Local-Global Principle) and the $p$-adic Schanuel Conjecture. Preliminary experiments with an implementation of this algorithm within the tool SKOLEM point to the practical applicability of this method.

Skolem's conjecture (roughly speaking) says that if an exponential Diophantine equation has no solution, than the equation has already no solution modulo $m$, with an appropriate $m$. In the talk we summarize some recent results justifying the conjecture for certain three term equations. The handled cases include Catalan's equation and Fermat's equation, for arbitrary, fixed bases. Note that previously Skolem's conjecture (in the integral case) was proved only for equations of the shape $a_1^{x_1}\cdots a_n^{x_n}=k$ by Schinzel.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider Diophantine equations with unknowns taken from finitely generated domains of characteristic $0$. Up to isomorphism, such a domain is of the shape $\mathbb Z[X_1,\ldots, X_r]/\mathcal I$, where $\mathcal I$ is a prime ideal of $\mathbb Z [X_1,\ldots, X_r]$ with $\mathcal I\cap\mathbb Z =(0)$. Special cases of such domains are rings of $(S-)$-integers in number fields and polynomial rings over $\mathbb Z$.

Lang (1960) was the first to prove finiteness results over arbitrary finitely generated domains of characteristic $0$ for various classes of Diophantine equations, by combining Roth's theorem over number fields, Roth's theorem over function fields, and specialization arguments. His finiteness results were ineffective, in that their proofs did not provide methods to determine all solutions. Győry (1983/84) proved $effective$ finiteness results for certain classes of Diophantine equations, valid for a restricted class of finitely generated domains. Later, Győry and E. (2013) managed to generalize Győry's effective method to arbitrary finitely generated domains of characteristic $0$. The idea is to map the equation over the finitely generated domain under consideration to various related equations over rings of $S$-integers in number fields by means of specializations and use effective height estimates for the solutions of the equations over the $S$-integers, e.g., obtained by means of Baker's method. Which specializations to use is controlled by height estimates for the solutions of related equations over function fields. In 2013, Győry and E. obtained in this manner an effective finiteness result for unit equations $ax+by=c$ in $x,y\in A^*$, with $A$ any finitely generated domain of characteristic $0$. This was extended later to various other classes of Diophantine equations over finitely generated domains.

In my talk I would like to give an idea how the method of Győry and E. works and give some applications. This is a prequel to the talk of Attila Bérczes.

Reference:
J.-H. Evertse, K. Győry: Effective results and methods for Diophantine equations over finitely generated domains, London Math. Soc. Lecture Note Ser. 475.

Let $A:={\mathbb Z}[z_1, \dots, z_r]\supset {\mathbb Z}$ be a finitely generated integral domain over ${\mathbb Z}$ and denote by $K$ the quotient field of $A$. Finiteness results for several kinds of Diophantine equations over $A$ date back to the middle of the last century. S. Lang generalized several earlier results on Diophantine equations over the integers to results over $A$, including results concerning unit equations, Thue-equations and integral points on curves. However, all his results were ineffective. The first effective results for Diophantine equations over finitely generated domains were published in the 1980's, when Győry developed his new effective specialization method. This enabled him to prove effective results over finitely generated domains of a special type. In 2011 Evertse and Győry refined the method of Győry such that they were able to prove effective results for unit equations $ax+by=1$ in $x,y\in A^*$ over arbitrary finitely generated domains $A$ of characteristic $0$. Using this new general method Bérczes, Evertse and Győry obtained effective results for Thue equations, hyper- and superelliptic equations and for the Schinzel-Tijdeman equation over arbitrary finitely generated domains. Koymans generalized the effective result of Tijdeman on the Catalan equation for finitely generated domains, while Evertse and Győry proved effective results for decomposable form equations in this generality. Bérczes proved effective results for equations $F(x,y)=0$ in $x,y\in A^*$ for arbitrary finitely generated domains $A$, and for $F(x,y)=0$ in $x,y\in \overline\Gamma$, where $F(X,Y)$ is a bivariate polynomial over $A$ and $\overline\Gamma$ is the division group of a finitely generated subgroup $\Gamma$ of $K^*$. In my talk I will focus mainly on these latter mentioned results, a short survey how the method of Evertse and Győry could be used in the proof of these results.

The $S$-unit equations in two unknowns, equations of the form $$\begin{align*} \alpha x+\beta y=1, \end{align*}$$ where the unknowns $x,y$ are $S$-units in a number field $K$ containing $\alpha,\beta$, are very important in the solution of many other families of Diophantine equations. For their application to obtaining the complete solution of Diophantine equations, an upper bound on the (height of) solutions of associated $S$-unit equations is required. The speaker (1974,79) gave explicit upper bounds for (slightly more general) solutions of $S$-unit equations, and used them to get various applications. Later several authors, including Evertse, Stewart, Tijdeman and Gy (1988), Bombieri (1993), Bugeaud and Gy (1996), Bugeaud (1998), Yu and Gy (2006) and Evertse and Gy (2015) improved upon or modified the previous bounds. Their bounds depend among others on the cardinality $|S|$ of $S$ and the largest norm $P$ of the prime ideals in $S$. In Yu and Gy (2006) we obtained two different, considerably improved bounds for the solutions. Le Fourn (2020) combined the proof of the first bound with his variant of Runge's method to replace $P$ in the first bound by the third largest norm $P'$ of the prime ideals in $S$. In Gy (2020) we refined the second, more complicated proof and combined it with Le Fourn's idea to replace $P$ in the second bound by $P'$. Further, we improved also the dependence on $|S|$ and, in terms of $S$, derived the best known bound to date for the solutions. In our talk we formulate the bounds from Gy (1979), Bugeaud and Gy (1996), Yu and Gy (2006), Le Fourn (2020), and Gy (2020), compare the bounds, emphasize the main tool and outline the main steps in the proof of Gy (2020). Further, we present some recent applications of our latest bound, giving improved upper bound on the $S$-integral solutions of Thue equations and some more general decomposable form equations over number fields, and providing the best Masser's type $ABC$ inequality to date towards Masser's $ABC$ conjecture over number fields; Gy (2022).

A theorem of Bilu and Tichy (2000) gives deep insight into the structure of polynomials $f,g$ with rational coefficients such that the equation $f(x) = g(y)$ has infinitely many rational solutions $x,y$. Lajos Hajdu and I have worked out the consequences in case $f$ has only simple rational roots, a case often considered in the literature. We describe the pairs $(deg(f), deg(g))$ which are possible, also in case both $f$ and $g$ have only simple rational roots. There is a connection with the classical Prouhet-Tarry-Escott problem to find two disjoint sets of $n$ integers such that the sums of the $k$-th powers are equal for $k$<$n$.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk we will study the negative Pell equation, which is the conic $C_D : x^2 - D y^2 = -1$ to be solved in integers $x, y \in \mathbb{Z}$. We shall be concerned with the following question: as we vary over squarefree integers $D$, how often is $C_D$ soluble? Stevenhagen conjectured an asymptotic formula for such $D$. Fouvry and Kluners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano.

We call a complex polynomial $D$ “pellian” if there are non-constant polynomials $A$ and $B$ such that $A^2-DB^2=1$. While all non-square quadratic polynomials are pellian, there are square-free polynomials of any even degree $\geq 4$ that are not pellian. Masser and Zannier considered one-parameter families of polynomials which are non-identically pellian and studied the pellian specialisations. They gave a criterion for the existence of infinitely many pellian specialisation. In joint work with Laura Capuano and Umberto Zannier we consider the “moduli space” of monic polynomials of fixed even degree $2d \geq 4$ and prove, among other things, that the locus of pellian polynomials consists of a denumerable union of subvarieties of dimension at most $d+1$.

Given n multiplicatively independent rational functions $f_1, \ldots, f_n$ with rational coefficients, there are at most finitely many complex numbers $a$ such that $f_1(a), \ldots, f_n(a)$ satisfy two independent multiplicative relations. This was proved independently by Maurin and by Bombieri, Habegger, Masser and Zannier, and it is an instance of more general conjectures of unlikely intersections over tori made by Bombieri, Masser and Zannier and independently by Zilber. We consider a positive characteristic variant of this problem, proving that, for sufficiently large primes, the cardinality of the set of $a \in \mathbb F_p$ such that $f_1(a), \ldots, f_n(a)$ satisfy two independent multiplicative relations with exponents bounded by a constant $K$ is bounded independently of $K$ and $p$. We prove also analogous results for products of elliptic curves and for split semiabelian varieties. This is a joint work with F. Barroero, L. Mérai, A. Ostafe and M. Sha.

In a series of papers, Sárközy and Stewart studied the prime divisors of sum-sets $\mathcal{A}+\mathcal{B}$. Among others, they showed that if $\mathcal{A},\mathcal{B}\subset \{1,\dots, N\}$ are not too small, then there are $a\in\mathcal{A}$ and $b\in\mathcal{B}$ such that $a+b$ has large prime divisors. In this talk we explore this problem for polynomials over finite fields. In particular, we show that if $\mathcal{A},\mathcal{B} \subset \mathbb{F}_q[x]$ are sets of polynomials of degree $n$, then $a+b$ has large degree irreducible divisors for some $a\in\mathcal{A}, b\in\mathcal{B}$. In particular, if $\mathcal{A},\mathcal{B}$ have positive relative densities, then $a+b$ has an irreducible divisor of degree $n+O(1)$ for some $a\in\mathcal{A}, b\in\mathcal{B}$.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I would like to give some new examples of curves in $E^n$, with $E$ an elliptic curve, for which we can give all rational points. These examples are interesting because the rank of the elliptic curve is larger than in other methods. These examples are related to a diophantine approximation method in the context of anomalous intersections.

Let $K$ be a number field with $\textrm{char}(K)\neq 2,3$ and let ${\mathcal{E}}$ be an elliptic curve defined over $K$. For every positive integer $m$, the $m$-division field $K({\mathcal{E}}[m])$ is the field generated over $K$ by the coordinates of the $m$-torsion points of ${\mathcal{E}}$. In the study of the arithmetic of elliptic curves, the fields $K({\mathcal{E}}[m])$ have played an important rôle. The investigation of the Galois representations on the total Tate module, Iwasawa theory, modularity and even the proof of the Mordell-Weil theorem are related to the properties of $K({\mathcal{E}}[m])/K$. When $m=p^r$, with $p\geq 5$ a prime and $r$ a positive integer, we prove $K({\mathcal{E}}[p^r])=K(x_1,\zeta_p,y_2)$, where $\{(x_1, y_1),(x_2,y_2)\}$ is a generating system of ${\mathcal{E}}[p^r]$ and $\zeta_p$ is a primitive $p$-th root of unity. In addition we produce an upper bound to the logarithmic height of the discriminant of the extension $K({\mathcal{E}}[m])/K$, for all $m\geq 3$. As a consequence, we give an effective version of the hypothesis of the following local-global divisibility problem in elliptic curves over number fields, where the local conditions are known only for finitely many places.

Problem(Dvornicich, Zannier, 2001). Let $M_K$ be the set of places $v\in K$ and let $K_v$ be the completion of $K$ at the valuation $v$. Suppose that for all but finitely many $v\in M_K$, there exists $D_v\in {\mathcal{E}}(K_v)$ such that $P=mD_v$, where $P$ is a fixed $K$-rational point of $\mathcal{E}$. Is it possible to conclude that there exists $D\in {\mathcal{E}}(K)$ such that $P=mD$?

This is a joint work with Roberto Dvornicich.

The classical Chabauty-Coleman theorem gives an explicit upper bound for the number of rational points on a hyperbolic curve by $p$-adic means. I'll explain an analogous result for surfaces. While the statement is completely analogous to the case of curves, the proof is rather different and it is based on the theory of $\omega$-integral curves and overdetermined systems of ODEs. This is joint work with Jerson Caro.

We estimate the growth rate of the function which counts the number of torsion points of order at most $T$ on an algebraic subvariety of the algebraic torus $\mathbb{G}_m^n$ over some algebraically closed field. We will see that there is a general upper bound which is sharp, and characterize the subvarieties for which the growth rate is maximal. For all other subvarieties there is a better bound which is power saving compared to the general one.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Let $n \ge 1$ be an integer and $\alpha_1, \ldots, \alpha_n$ be non-zero algebraic numbers. Let $b_1, \ldots , b_n$ be integers with $b_n \not= 0$, and set $B = \max\{3, |b_1|, \ldots , |b_n|\}$. For $j =1, \ldots, n$, set $h^* (\alpha_j) = \max\{h(\alpha_j), 2\}$, where $h$ denotes the (logarithmic) Weil height. Assume that the quantity $\Lambda = b_1 \log \alpha_1 + \cdots + b_n \log \alpha_n$ is nonzero. A typical lower bound of $\log |\Lambda|$ given by Baker's theory of linear forms in logarithms takes the shape $$- c(n, D) \, h^* (\alpha_1) \ldots h^*(\alpha_n) \log B,$$ where $c(n,D)$ is positive, effectively computable and depends only on $n$ and on the degree $D$ of the field generated by $\alpha_1, \ldots , \alpha_n$. However, in certain special cases and in particular when $|b_n| = 1$, this bound can be improved to $$- c(n, D) \, h^* (\alpha_1) \ldots h^*(\alpha_n) \log \frac{B}{h^*(\alpha_n)}.$$ The term $B' := B / h^*(\alpha_n)$ in place of $B$ originates in works of Feldman and of Baker. It is a key tool for improving, in an effective way, the upper bound for the irrationality exponent of a real algebraic number of degree at least $3$ given by Liouville's theorem. We survey various applications of this $B'$ to exponents of approximation evaluated at algebraic numbers, to the $S$-part of integer sequences, and to Diophantine equations.

By a result of Zimmert (1981) the regulator of a number field $K$ grows at least exponentially with the degree of $K$. The regulator is closely related to the (co)volume (of the image via the logarithmic embedding) of the full lattice of units of $K$. Thus a natural question concerns the (co)volume of subgroups of the group of units. For a one dimensional subgroup this question turns out to be equivalent to Lehmer's problem on the height of algebraic numbers. Bertrand and Rodriguez-Villegas independently formulate conjectures which interpolate between one dimensional and full dimensional subgroups. We discuss some recent results on these conjectures.

The genus group of an extension $K/k$ of number fields is a certain natural quotient of the class group of $K$. We discuss the average cardinality of the genus group, as $K$ ranges over Galois extensions of $k$ with fixed abelian Galois group, ordered by conductor. This is joint work with Dan Loughran and Rachel Newton.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will present new results on Malle's conjecture for nilpotent groups, a joint work with Koymans. I will relate this work to the problem of lower bounding arboreal degrees, and report ongoing work in progress with Mello, Ostafe and Shparlinski, along with past work of mine on the subject. I will finally relate these works with joint work with Ferraguti on (abelian) arboreal Galois representations.

For a large class of integer polynomials, we link irreducibility of their iterates modulo a prime $p$ (also known as dynamical irreducibility) to the distribution of quadratic residue and non-residues in certain specialisations of their iterates. This allows us to use bounds of character sums to study dynamical irreducibility for almost all $p$. There are however some Diophantine obstacles related to the possible existence of many squares in the above specialisations when considered over $\mathbb Q$. We will explain this obstacle and discuss possible ways to overcome it (which also work for quadratic polynomials and some special trinomials). Joint work with Laszlo Merai and Alina Ostafe

In most circumstances, proving estimates better than those tantamount to square-root cancellation for mean values of exponential sums remains a distant prospect. It is classical that this is possible for small moments of quadratic Weyl sums. In this talk, we describe progress for higher degree exponential sums associated with Vinogradov’s mean value theorem. It transpires that a natural extension of the Main Conjecture in Vinogradov’s mean value theorem delivers subconvex estimates for twisted moments at the critical exponent, and that such conclusions may be proved unconditionally in the cubic case.

The (renormalized) gap distribution is a popular statistic for studying how random a deterministic sequence really is. While the gap distribution of many classical sequences is conjectured to be a Poisson distribution, there are hardly any examples known for which this can be (dis)proven! One such exception is $(\sqrt{n}\,\mathrm{mod\,}1)_{n\geq1}$. In the 2000's, Elkies and Mcmullen showed that the gap distribution of $(\sqrt{n}\,\mathrm{mod\,}1)_{n\geq1}$ exists and is $not$ a Poisson distribution. Their (ineffective) proof relies Teichmüller theory and homogeneous dynamics, in particular on Ratner's theorem. Some years ago, Browning and Vinogradov made the proof of Elkies and Mcmullen effective. In a recent work with Maksym Radziwiłł, we give a rather different (effective) proof, relying on purely analytic methods, which one could even describe as elementary. We shall discuss the basic strategy of this new approach.

In 1980, Mohanty conjectured that a non-trivial arithmetic progression of rational points on a Mordell elliptic curve cannot have more than four terms. In earlier joint work with Hector Pasten, we proved that the maximal length of a non-trivial arithmetic progression on an elliptic curve only depends on its rank and its j-invariant, hence unconditionally proving a version of Mohanty's conjecture for several families of elliptic curves. One can study geometric progressions on elliptic curves and try to find a bound of the maximal length of non-trivial geometric progressions, depending on similar data. The case of geometric progressions, however, turns out to be much more delicate from a technical point of view and new ideas are necessary. In this talk I will show how to get a bound of this type for geometric progressions on elliptic curves.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We will discuss a quantitative bound for the number of points of small Néron-Tate height in the embedding of a curve over a function field into its Jacobian. The proof uses Zhang's admissible pairing on curves, the arithmetic Hodge index theorem over function fields, and the metrized graph analogue of Elkies’€™ lower bound for the Green function. As a special case, we will show that the number of torsion points on the curve is bounded by $16g^2+32g+124$, where g denotes the genus. This is joint work with Nicole Looper and Joseph Silverman.

Iterations of rational maps on the projective line are ubiquitous in mathematics and appear in number theory as well as numerical analysis, for example in the Newton method. I will talk mainly about joint work with Mavraki in which we study families of rational maps and their specialisation maps. Our main goal is to understand properties that hold uniformly for all specialisations. Our investigations have connections to a relative Bogomolov conjecture for dynamical systems and use tools such as heights and equi-distribution.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

My plan is to give a more detailed construction of Diophantine approximation sets for properly intersecting nonzero and effective Cartier divisors on a given polarized projective variety. I will then outline a proof of compactness of such approximation sets. This will expand on what I recently described, briefly, at BIRS this past June 2022. Also, I intend to survey some key concepts that allow for effective approaches for working with local Weil and logarithmic height functions.

Let $A_1, A_2\in \mathbb C(z)$ be rational functions of degree at least two that are neither Latt\`es maps nor conjugate to $z^{\pm n}$ or $\pm T_n.$ In the talk, we describe invariant, periodic, and preperiodic algebraic curves for endomorphisms of $\mathbb P^1\times \mathbb P^1$ of the form $(z_1,z_2)\rightarrow (A_1(z_1),A_2(z_2)).$ In particular, we show that if $A\in \mathbb C (z)$ is not a ``generalized Lattés map'', then any $(A,A)$-invariant curve has genus zero and can be parametrized by rational functions commuting with $A$. As an application, for $A$ defined over a number field we give a criterion for a point of $\mathbb P^1\times \mathbb P^1$ to have a Zariski dense $(A, A)$-orbit in terms of canonical heights, and deduce from this criterion a version of a conjecture of Zhang.