### Xiamen Workshop in Diophantine Equations and Related Topics

Xiamen Workshop in Diophantine   Equations and Related Topics

April 22Room 105, Laboratory Building, Haiyun Campus

 09:00-09:20 Opening Ceremony, Photos 09:20-10:00 Fei XU: Recent   progress on strong approximation with Brauer-Manin obstruction 10:00-10:30 Tea Break 10:30-11:10 Kai-Man TSANG: Waring-Goldbach problem for cubes 11:20-12:00 Liang-Chung HSIA: On a GCD problem for iterates   of polynomials 14:00-14:40 Chieh-Yu CHANG: Computing the dimensions of   double zeta values in positive characteristic 14:50-15:30 Guangshi LU: Mean value theorem connected with Fourier   coefficients of automorphic forms for higher rank groups 15:30-16:00 Tea Break 16:00-16:40 Qiming YAN: Moving   Target Problems in Nevanlinna theory and Diophantine approximation 16:50-17:30 Preda MIHAILESCU: On the error term of the Dedekind zeta function

April 23Room 105, Laboratory Building, Haiyun Campus

 09:20-10:00 Lilu ZHAO: On translation invariant quadratic forms in dense sets 10:00-10:30 Tea Break 10:30-11:10 Pingzhi YUAN: Je´ smanowicz’ conjecture and related equations 11:20-12:00 Haigang ZHOU: Certain   Diophantine problems and Jacobi forms 14:00-14:40 Dasheng WEI: Integral   points for a variety with a group action 14:50-15:30 Bo HE: On Diophantine quintuple   conjecture 15:30-16:00 Tea Break 16:00-16:40 Ronggang SHI: The   distribution of lattice points given by Dirichet’s theorem.

Abstracts

April  22 Saturday

Abstract: In this talk, I'll give a survey report of recent progress on strong approximation with Brauer-Manin obstruction for family of homogeneous spaces, partial equivariant compactifications of connected linear algebraic groups and homogeneous spaces by fibration method, the classical descent method, the descent method involving Brauer-Manin obstruction and so on.

Abstract: A well-known and difficulty conjecture for cubes of primes is that every sufficiently large integer satisfying certain necessary congruence conditions is the sum of four cubes of primes numbers. Hua showed in 1965 that nine cubes of primes are sufficient and this remains the best of to date.  It has also been shown that various combinations of eight cubes of almost primes are also sufficient.  In this talk we shall discuss some results related to this problem.  This is a joint work with Tak-Wing Ching.

Abstract: Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have $\gcd(a^n - 1, b^n - 1) \mid h$.  In this talk, we'll discuss a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$ are nonconstant "compositionally independent" polynomials and $c(x) \in {\mathbb C}[x]$, then there are at most finitely many $\lambda\in {\mathbb C}$ with the property that there is an $n$ such that $(x - \lambda)$ divides  $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$.This is a joint work with Tom Tucker.

Abstract: In this talk, we will present an effective algorithm for computing the dimensions of double zeta values in positive characteristic. We will show how a variant of Siegel's lemma plays a key role in the design.

Abstract: In this talk, I shall introduce some results on the distribution of Fourier coefficients of automorphic forms for higher rank groups. In particular, I will talk about the orthogonality between additive characters and Fourier coefficients over primes, and Bombieri-Vinogradovt ype mean value connected with Fourier coefficients. These results are related to the Möbius randomness principle.

Abstract: In 1985, Osgood proved Nevanlinna's conjecture for moving targets (motivated, surprisingly, by the proof of Roth's theorem and Osgood's own analogy with Nevanlinna theory). Due to Osgood's observation and subsequent work done by S. Lang, P. Vojta and others, it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) and theory of Diophantine approximation may be somehow related.  In this talk, we give a partial survey of the development on moving target problems in Nevanlinna theory and Diophantine approximation.

Preda MIHAILESCU : On the error term of the Dedekind zeta function.

Abstract:The proof of convergence and the evaluation of the residue of the Dedekind zeta function at the value s = 1 are exemplary deep and beautiful applications of the geometry of numbers and the ideas established by Minkowski. Unfortunately, if one is also interested in the error term, the text books and the classical results all stop with mentioning the order of magnitude, based on some unspecified Lipschitz constants. Even more, a folklore point of view used to suggest that one should expect that the constant in the error term should be exponential in the regulator, thus Large!

Recently, Korneel Debaene undertook the generalization of an ideea used by Wolfgang Schmidt in the case of quadratic fields, and thus proved a very well behaved constant factor for the error term, in which the regulator explicitly only occurs at most linearly. However, some cofactors involved so high powers of the extension degree, that in certain cases of interest in Iwasawa theory, these constants became comparable to exponentials in the regulator. Having contributed to the improvement of those constants, thus reaching an unconditional at most linear behavior in the constants, I shall expose in this talk the fundamental ideas of these quite technical improvements.

April  23 Sunday

Abstract: Let $f(x_1,\ldots,x_n)$ be a translation invariant indefinite quadratic form with $n\ge 9$ and let $N$ be sufficiently large. We prove there exists $c_f>0$ such that $f(x_1,\ldots,x_n)=0$ has nontrivial zeros in $\mathcal{A} \subseteq \{1,2,\ldots,N\}$ as soon as $|\mathcal{A}|\ge N(\log N)^{-c_f}$.

Abstract: Let $(a, b, c)$ be a primitive Pythagorean triple such that $a^2 +b^2 = c^2$ and $2|b$. Let $n$ be a positive integer. In 1956, L. Jesmanowicz conjectured the equation $(an)^x + (bn)^y = (cn)^z$ has only the positive integral solution $(x, y, z) = (2, 2, 2)$ for any $n$. This conjecture has been proved to be true in many special cases for $n = 1$. But, in general, the problem is not solved as yet. When $n > 1$, only a few results on this conjecture are known. In this talk, we will talk about some results on the conjecture and related Diophantine equations.

Abstract: We use the theory of Jacobi forms to study the number of elements in a maximal order of a definite quaternion algebra over the field of rational numbers whose characteristic polynomial equals a given polynomial. A certain weighted average of such numbers equals (up to some trivial factors) the Hurwitz class number $H(4n-r^2)$. As applications, we can obtain many explicit formulas for the number of solutions of Diophantine equations systems.



Abstract: Let k be a number field and X a smooth integral variety with a surjective morphism from X to the affine line. Assume that its generic fiber is a homogeneous space of a simply connected semisimple group. We will discuss the existence of integral points of the variety X. It's a joint work with Fei Xu.

何波Bo HE: On Diophantine quintuple conjecture.

Abstract: A set of $m$ positive integers $\{a_1, a_2, \dots ,a_m\}$ is called a Diophantine $m$-tuple if $a_i a_j + 1$ is a perfect square for all $1 \le i< j \le m$. In 2004, Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$-tuples states that no Diophantine quintuple exists at all. In this talk we show how to prove this conjecture. This is a joint work with Alain Togbe and Volker Ziegler.

Abstract: The Dirichlet’s theorem in Diophantine approximation is about the approximation of vectors in an n-dimensional Euclidean space by rational vectors. Each rational vector corresponds to an integral vector in the (n+1)-dimensional Euclidean space. Given a generic point of the n-dimensional Euclidean space, we count the integral vectors satisfying the Dirichlet’s theorem with fixed directions. The counting uses pointwise equidistribution for one parameter diagonal group action on homogeneous space with respect to unbounded function. This result is a joint work with Dmitry Kleinbock and Barak Weiss.