Titles and abstracts

Transcription

1 Titles and abstracts Workshop Effective Methods for Diophantine Problems *** Mike Bennett (University of British Columbia) An old and new approach to Goormaghtigh s equation This is joint work with Adela Gherga and Dijana Kreso. I will discuss an attempt to solve a classical polynomial-exponential Diophantine equation via machinery based upon off-diagonal Padé approximation. Our main result is that the equation x m 1 x 1 = yn 1 y 1 has, for integers m > n > 2 and y > x > 1, with gcd(m 1, n 1) > 1, at most finitely many solutions with n fixed. Csanád Bertók (University of Debrecen) Solving exponential Diophantine equations by using local methods By using Baker s theory it is possible to derive upper bounds for the size of the solutions of a wide family of Diophantine equations. However when the number of terms in an equation is large then this method cannot be used. For some special cases we may use other tools to achieve bounds for the number of solutions, but these methods also fail when we start to increase the number of terms. In this short talk we will present an algorithm and theoretical results which can be used to find all solutions of a wide variety of exponential Diophantine equations. We will illustrate our method by examples and numerical results as well. Yuri Bilu (Université Bordeaux) A j-invariant of a CM elliptic curve is not a unit A singular modulus is a j-invariant of an elliptic curve with Complex Multiplication. It is known since the 19th century that any singular modulus is an algebraic integer. Masser asked in 2011 whether only finitely many of them are units (invertible elements in the ring of all algebraic integers); call them singular units. Habegger (2015) answered this positively, proving that there exist at most finitely many singular units. However, his argument was non-effective (dependent on Siegel s zero through using the Duke Equidistribution Theorem), and did not provide any upper bound for the 1

2 size of these singular units. I will report on a recent joint work with Philipp Habegger and Lars Kühne, where we prove that singular units do not exist at all. First, we show that the absolute value of the discriminant of any singular unit is bounded by Next, we use computer-assisted arguments to rule out singular units with discriminants smaller than this bound. Yann Bugeaud (Université Strasbourg) Polynomial root separation We discuss how close the nonzero difference between two roots of an irreducible, integer polynomial can be in terms of its degree and the absolute values of its coefficients. Let d 2 be an integer and e(d) the infimum of the real numbers δ for which α i α j H(P ) δ 1 i<j k holds for every integer polynomial P (X) of degree d and sufficiently large height H(P ), with distinct roots α 1,..., α d. In 1964, Mahler established that e(d) d 1. It is easy to see that e(2) = 1. Evertse (2004) and, independently and by means of a simpler approach, Schönhage (2006) established that e(3) = 2. However, the exact value of e(d) for d 4 remains unknown: it has been established by Bugeaud and Dujella (2011) that e(d) exceeds d/2, but we can even not rule out that e(d) = d 1. We survey recent results on this and related questions. Sander Dahmen (VU Amsterdam) Some modular and related methods We will discuss several aspects of modular methods and related/analogous methods, such as (hyper)elliptic Frey curve constructions and number field enumerations. These will be illustrated by (sometimes partly) solving explicit Diophantine problems, including classical ones like Thue-Mahler equations and generalized Fermat equations, as well as other types. Finally, we will also briefly touch upon a somewhat different theme, namely the automation and formalization of proofs in this setting. 2

3 Netan Dogra (Imperial College, London) Unlikely intersections and the Chabauty-Kim method in higher dimensions A crucial input in Chabauty s method for curves over Q (and in its generalisation due to Kim) is the finiteness of zeroes of p-adic (iterated) integrals on curves. To extend the Chabauty-Kim method to number fields and to higher dimensional varieties, one must control the zeroes of iterated integrals on higher dimensional varieties. In my talk I will explain an unlikely intersection result for the zeroes of such functions and its Diophantine applications. Kálmán Győry (University of Debrecen) 35 years of collaboration with Jan-Hendrik Evertse In my lecture some old and new joint results will be presented from the following topics: (1) Decomposable form equations and unit equations (their equivalence, finiteness criteria, quantitative and effective results, applications), (2) Discriminant equations (quantitative and effective results, applications), (3) Effective finiteness results for Diophantine equations over finitely generated domains. Ariyan Javanpeykar (Universität Mainz) Arithmetic hyperbolicity A variety is arithmetically hyperbolic if it has only finitely many integral points. What properties do (or should) such varieties have? Lang s conjecture predicts for instance that such varieties have a finite automorphism group. We will explain how one can prove that projective arithmetically hyperbolic varieties have only finitely many automorphisms. Rafael von Känel (Princeton University) Solving cubic Thue equations via Shimura-Taniyama conjecture In this talk we discuss a practical method which allows to find all S-integral solutions of cubic Thue equations (over Q). After explaining the general strategy which combines the method of Faltings (Arakelov, Parshin, Szpiro) with the Shimura-Taniyama conjecture, we consider some aspects of the elliptic logarithm sieve used in our method. This is joint work with Benjamin Matschke. 3

4 Makoto Kawashima (Osaka University) A lower bound for linear forms in logarithms of algebraic numbers via Padé approximation Let K be an algebraic number field. For (α 1,..., α m ) K m of sufficiently small absolute value, we present a new lower bound for the absolute value of linear forms in 1, log(1 α 1 ),..., log(1 α m ) with rational coefficients, by means of Hermite-Padé approximation. We give results both in the complex case and the p-adic case. This is a joint work with Noriko Hirata-Kohno. Arno Kret (University of Amsterdam) Integral points on Hilbert modular varieties We will discuss a joint project with Rafael von Känel in which we studied integral points on Hilbert modular varieties. In particular we shall present explicit upper bounds for the height and number of integral points on Hilbert modular varieties. We shall also explain the strategy of proof. Florian Luca (University of Witwatersrand/University of Ostrava) X-coordinates of Pell equations in various sequences Let d > 1 be a squarefree integer and (X n, Y n ) be the nth solution of the Pell equation X 2 dy 2 = ±1. Given your favourite set of positive integers U, one can ask what can we say about those d such that X n U for some n? Formulated in this way, the question has many solutions d since one can always pick u U and write u 2 ± 1 = dv 2 with integers d and v such that d is squarefree obtaining in this way that (u, v) is a solution of the Pell equation corresponding to d. What about if we ask that X n U for at least two different n s? Then the answer is very different. For example, if U is the set of squares, then it is a classical result of Ljunggren that the only such d is 1785 for which both X 1 and X 2 are squares. In my talk, I will survey recent results about this problem when U is the set of Fibonacci numbers, Tribonacci numbers, k-generalized Fibonacci numbers, sums of two Fibonacci numbers, rep-digits (in base 10 or any integer base b 2), and factorials. The proofs use linear forms in logarithms and computations. These results have been obtained in joint work with various colleagues such as J. J. Bravo, C. A. Gómez, A. Montejano, L. Szalay and A. Togbé and recent Ph.D. students M. Ddamulira, B. Faye and M. Sias. 4

5 Benjamin Matschke (Koç University) Solving cubic Thue Mahler equations via the Shimura Taniyama conjecture In this talk we present a practical algorithm to solve cubic Thue Mahler equations. Our algorithm relies on new height bounds, which we obtained using the method of Faltings (Arakelov, Parshin, Szpiro) combined with the Shimura Taniyama conjecture (without relying on lower bounds on linear forms in logarithms), as well as several improved and new sieves. As an application, we computed the solutions of large classes of Thue Mahler equations. Evertse proved an upper bound for the number of solutions for general Thue Mahler equations of arbitrary degree over number fields. We used our resulting data to motivate conjectures and questions on the number of solutions, also for associated diophantine problems. This is joint work with Rafael von Känel. Héctor Pastén (Harvard University) Shimura curves and the abc conjecture I will explain some new connections between the abc conjecture and modular forms. In particular, I will outline a proof of a new unconditional estimate for the abc conjecture: the product of the p-adic valuations of an abc triple (p varying over the factors of abc) is bounded polynomially in the radical of abc. Similarly, the fudge factor of elliptic curves is bounded polynomially in the conductor. The proofs involve a number of tools such as Shimura curves, CM points, analytic number theory, and Arakelov geometry. Some intermediate results are of independent interest, such as bounds for the Manin constant beyond the semi-stable case. If time permits, I will also explain some results towards Szpiro s conjecture over totally real number fields which are compatible with the discriminant term appearing in Vojta s conjecture for algebraic points of bounded degree. István Pink (University of Debrecen) Some applications of Baker s method to Diophantine equations In this talk we will concentrate on some applications of the famous method of Baker to Diophantine equations. After a concise introduction to the theory of linear forms in logarithms of algebraic numbers we give a brief survey on some classical results in Diophantine number theory involving Baker s method. Then we point out several refinements of the original work of 5

6 Baker and we indicate how the combination of these refinements with some other effective methods (e.g. LLL algorithm, local method, hypergeometric method) may lead to effective resolution of Diophantine equations of certain type. Marusia Rebolledo (Université Blaise Pascal Clermont-Ferrand) A moduli interpretation for the non-split Cartan modular curves Here, we are interested in the modular curves associated to non-split Cartan subgroups or their normalizer in GL 2 (F p ). These modular curves appear for instance in Serre s problem of classifying all possible Galois structures of p-torsion points on elliptic curves over number fields. With Christian Wuthrich, we proposed a description of those curves as moduli spaces, namely classifying elliptic curves endowed with a level structure that we call a necklace. I will show how this description allows to recover some classical results (on elliptic points, degenerate maps, Hecke operators etc.) as well as gives a more explicit and geometric version of a theorem of Chen. László Remete (University of Debrecen) Thue equations and monogenity of algebraic number fields An algebraic number field K of degree n is called monogene if its ring of integers Z K is a simple ring extension of Z, that is Z K = Z[α]. In this case {1, α,..., α n 1 } is an integral basis of K called power integral basis. To decide if a number field is monogene and to determine all generators of power integral bases are classical problems of algebraic number theory. These problems lead to special types of Diophantine equations, called index form equations, that can often be reduced to various types of Thue equations. Therefore the effective methods for solving Thue equations and related equations can be very well applied in monogenity problems. Starting from cubic and quartic number fields we describe several results on monogenity of number fields, involving some very recent results on infinite parametric families of pure fields and simplest sextic fields. Tarlok Shorey (Institute for Advanced Study, Bangalore) Baker s explicit abc-conjecture Alan Baker in 2004 formulated an explicit version of the abc-conjecture. I shall speak on it together with its recent applications. 6

7 Samir Siksek (Warwick University) Frey curves, short character sums and a problem of Erdős Consider the following Diophantine problem: n(n + d)(n + 2d) (n + (k 1)d) = y l, gcd(n, d) = 1, where n, d, y are integers and the exponent l is prime. There are obvious solutions with y = 0 or d = 0. A long-standing conjecture of Erdős states that if k is suitably large then the only solutions are the obvious ones. We show that if k is suitably large then either the solution is one of the obvious ones, or l < exp(10 k ). Our methods include Frey curves and Galois representations, the prime number theorem for Dirichlet characters, and results on exceptional zeros of Dirichlet L-functions. This is joint work with Mike Bennett. Cameron Stewart (University of Waterloo) On the number of solutions of S-unit equations In this talk we shall discuss results obtained on the number of solutions of S-unit equations and related Diophantine equations. In particular we shall focus on the work of Jan-Hendrik Evertse in this context. Rob Tijdeman (University of Leiden) On the equation ax n + by n = c and its generalizations The title equation is basic for several classes of Diophantine equations such as S-unit equations, Thue(-Mahler) equations and values of linear recurrence sequences. Here we assume that a, b, c are fixed nonzero rational integers and n 3. In 1909 Thue proved that the number of solutions x, y Z is finite. Siegel showed in 1937 that if ab is sufficiently large compared to c, there is at most one solution (x, y). The Gelfond-Baker method implies that n is bounded. In his thesis Evertse used an improvement of Siegel s result to give an upper bound for the number of solutions depending only on n and c. In the lecture the history of the title equation and some of its generalizations will be summarized. 7