11 Algophant 2017

# Journées Algophantiennes Bordelaises 2017 June 7—9, Université de Bordeaux

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### Abstracts

New: Some speakers kindly provided us with their slides.

Samuele Anni
Diophantine equations and semistable elliptic curves over totally real fields
(slides)
In this talk I will show that the generalized Fermat equation $x^{2\ell}+y^{2 m}=z^p$ has no non-trivial primitive solutions for primes $$\ell, m \ge 7$$, and $$3 \le p \le 13$$. This is achieved by relating a putative solution to a Frey curve over a real subfield of the $$p$$-th cyclotomic field, and studying its mod $$\ell$$ representation using modularity and level lowering.

In particular, I will describe, on the one hand, the modularity theorem for semistable elliptic curves over totally real number field used and, on the other hand, the computation with Hilbert modular forms involved.

This is joint work with Samir Siksek.

Jennifer S. Balakrishnan
p-adic heights and rational points on curves
(slides)
I will describe how $$p$$-adic heights can be used to find rational or integral points on certain curves of genus at least $$2$$. This is based on joint work with Amnon Besser, Netan Dogra, and Steffen Müller.

Kwok Chi Chim
A lower bound for linear forms in two p-adic logarithms
In this talk I will present a recent result on linear forms in two $$p$$-adic logarithms, where we establish an upper bound for the $$p$$-adic valuation $$v_p(\alpha_1^{b_1} - \alpha_2^{b_2})$$, where $$\alpha_1$$, $$\alpha_2$$ are algebraic numbers and $$b_1$$, $$b_2$$ are positive rational integers. In particular, the bound has a dependence on $$B$$ which relates with the logarithm of $$b_1$$ and $$b_2$$.

David Corwin
Motivic Periods and Coleman Functions
(slides)
I will discuss ongoing work with Ishai Dan-Cohen (continuing work such as https://arxiv.org/abs/1311.7008) in which we're using motivic periods to find Coleman functions that vanish on $$X(\mathbb{Z}[1/6])$$ where $$X$$ is the projective line minus three points.

Power of Two as Sums of Three Pell Numbers
(slides)
There are many papers in the literature dealing with Diophantine equations obtained by asking that members of some fixed binary recurrence sequence be squares, factorials, triangular, or belonging to some other interesting sequence of positive integers. In this talk, I will show all the solutions of the Diophantine equation $$P_\ell + P_m +P_n=2^a$$, in nonnegative integer variables $$(n,m,\ell, a)$$ where $$P_k$$ is the $$k$$-th term of the Pell sequence $$\{P_n\}_{n\ge 0}$$ given by $$P_0=0$$, $$P_1=1$$ and $$P_{n+1}=2P_{n}+ P_{n-1}$$ for all $$n\geq 1$$.

This is joint work with F. Luca and J. J. Bravo.

Akinari Hoshi
Cubic Thue equations and simplest cubic fields
I would like to give a talk which aims to explain the result of my paper "On correspondence between solutions of a family of cubic Thue equations and isomorphism classes of the simplest cubic fields" (J. Number Theory, 2011). This paper established "the field isomorphism method" to solve a family of Thue equations. This method enables us to determine all of the integer solutions to a family of Thue equations by using information of overlaps of their splitting fields. The main result can be obtained by using R. Okazaki's result on Geometric Gap Principle (2002) and some computer calculations.

Rafael von Känel
On Mordell equations
In this talk we will discuss joint work with Benjamin Matschke in which we solved in particular large classes of Mordell equations. After explaining the general strategy used to solve the equations, we will consider various questions motivated by our data.

Matija Kazalicki
Supersingular zeros of divisor polynomials of elliptic curves of prime conductor
For a prime number $$p$$ we study the zeros modulo $$p$$ of divisor polynomials of elliptic curves $$E/\mathbb{Q}$$ of conductor $$p$$. Ono made the observation that these zeros of are often $$j$$-invariants of supersingular elliptic curves over $$\overline{\mathbb{F}_{p}}$$. We show that these supersingular zeros are in bijection with zeros modulo $$p$$ of an associated quaternionic modular form $$v_E$$.

This allows us to prove that if the root number of $$E$$ is $$-1$$ then all supersingular $$j$$-invariants of elliptic curves defined over $$\mathbb{F}_{p}$$ are zeros of the corresponding divisor polynomial.

If the root number is $$1$$ we study the discrepancy between rank $$0$$ and higher rank elliptic curves, as in the latter case the amount of supersingular zeros in $$\mathbb{F}_p$$ seems to be larger. In order to partially explain this phenomenon, we conjecture that when $$E$$ has positive rank the values of the coefficients of $$v_E$$ corresponding to supersingular elliptic curves defined over $$\mathbb{F}_p$$ are even. We prove this conjecture in the case when the discriminant of $$E$$ is positive.

This is joint work with Daniel Kohen.

Hendrik Lenstra
Solving equations in orders
We consider the algorithmic problem of finding all zeroes of a monic polynomial in one variable with integer coefficients, in a commutative ring of which the additive group is a finitely generated free abelian group. In addition to several good algorithms in special cases, we present evidence that in its full generality the problem does not admit an efficient solution, even when the ring has a zero nilradical. The talk is partly based on joint work with Alice Silverberg.

Florian Luca
On Diophantine quadruples of Fibonacci numbers
(slides)
A Diophantine $$k$$-tuple is a set of $$k$$-positive integers $$\{a_1,\ldots, a_k\}$$ such that $$a_ia_j+1$$ is a square for all $$1\le i < j\le k$$. Recently, Bo He, Togbé and Ziegler proved that $$k\le 4$$. If $$(F_n)_{n\ge 0}$$ denotes the Fibonacci sequence given by $$F_0=0,~F_1=1$$ and $$F_{n+2}=F_{n+1}+F_n$$ for all $$n\ge 0$$, then $$\{F_{2n},F_{2n+2},F_{2n+4}\}$$ is a Diophantine triple for all $$n\ge 1$$. In my talk, I will show that there only finitely many Diophantine quadruples consisting of Fibonacci numbers. This is joint work with Y. Fujita.

Jean-François Mestre
Isogénies entre jacobiennes de courbes hyperelliptiques
L'algorithme de la moyenne arithmético-géométrique, dû à Legendre et Gauss, peut être interprété à l'aide de tours convenables d'isogénies de degré $$2$$ entre courbes elliptiques. Richelot (1836) a généralisé cet algorithme au cas des courbes de genre $$2$$. On montre ici que, pour tout entier $$g$$, il existe une infinité de couples de courbes hyperelliptiques dont les jacobiennes sont reliées par une isogénie de noyau cyclique d'ordre $$2^g$$, et on décrit dans le cas du genre $$3$$ la structure de tels couples de courbes.

Références:
- J-F. Mestre, J. Algebraic Geom. 22 (2013), 575-580, "Une généralisation d'une construction de Richelot".
- Ivan Boyer, Thèse, Paris-Diderot, Décembre 2013, "Variétés abéliennes et jacobiennes de courbes hyperelliptiques, en particulier à multiplication réelle ou complexe".

Steffen Müller
Quadratic Chabauty for hyperelliptic curves with RM Jacobian
I will discuss joint work with J. Balakrishnan and N. Dogra on the computation of rational points on curves of genus $$2$$ whose Jacobian has rank $$2$$ and real multiplication. The method is based on recent work of Balakrishnan and Dogra which links Kim's non-abelian Chabauty method to $$p$$-adic heights, enabling us to explicitly compute a finite set of $$p$$-adic points containing the rational points.

Filip Najman
Torsion of elliptic curves over number fields
(slides)
We will give an overview of known results about the torsion of elliptic curves over number fields, focusing on recent developments in the subject, and sketch the methods used to prove these results.

Vandita Patel
Sums of consecutive perfect powers is seldom a perfect power
(slides)
Let $$k$$ be an even integer such that $$k$$ is at least $$2$$. I would be happy to speak about a (natural) density result and show that for almost all $$d$$ at least $$2$$, the equation $$(x+1)^k+(x+2)^k+\ldots+(x+d)^k=y^n$$ with $$n$$ at least $$2$$, has no integer solutions $$(x,y,n)$$.

Samir Siksek
Frey Curves, the Large Sieve and a Problem of Erdős
Consider the following Diophantine problem: $n(n+d)(n+2d)\cdots (n+(k-1)d)=y^\ell, \qquad \gcd(n,d)=1,$ where $$n$$, $$d$$, $$y$$ are integers and the exponent $$\ell$$ 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 $$\ell<\exp(10^k)$$. Our methods include Frey curves and Galois representations, the prime number theorem for Dirichlet characters, results on exceptional zeros of Dirichlet $$L$$-functions, the large sieve, and Roth-like theorems on the existence of $$3$$-term arithmetic progressions in certain sets. This is joint work with Mike Bennett.

Michael Stoll
Rational points on curves in practice
I will give an overview of a range of methods that can be used to determine the set of rational (or also integral) points on a curve of genus at least $$2$$, explaining what is and what is not possible in practice.

George Turcas
Fermat's Last Theorem over Quadratic Imaginary Fields of Class Number one. An overview of the difficulties.
Assuming two deep but standard conjectures from the Langlands Programme, Sengun and Siksek proved that asymptotic Fermat's Last Theorem holds for imaginary quadratic fields $$\mathbb{Q}(\sqrt{-d})$$ with $$-d=2, 3 \mod 4$$. In this talk I will present how, under the assumption of the same two conjectures, we are trying to prove explicitly FLT over quadratic imaginary fields of class number $$1$$.

Francesco Veneziano
Unlikely intersections and the effective Mordell–Lang conjecture
I will recall the link between the subject of Unlikely Intersections and the effective Mordell–Lang Conjecture. In this context, I will present an explicit height bound for rational points on some curves and show many concrete examples in which it is possible to apply this result and list all such points.