Journées Algophantiennes Bordelaises 2017June 7—9, Université de Bordeaux  
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AbstractsNew: Some speakers kindly provided us with their slides.
Diophantine equations and semistable elliptic curves over totally real fields
In this talk I will show that the generalized Fermat equation \[x^{2\ell}+y^{2 m}=z^p\] has no nontrivial 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.
(slides) 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.
padic heights and rational points on curves
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.
(slides)
A lower bound for linear forms in two padic 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 \(b_2\).
Motivic Periods and Coleman Functions
I will discuss ongoing work with Ishai DanCohen (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.
(slides)
Power of Two as Sums of Three Pell Numbers
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 \(P_{n+1}=2P_{n}+ P_{n1}\) for all \(n\geq 1\).
(slides) This is joint work with F. Luca and J. J. Bravo.
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.
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.
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.
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.
On Diophantine quadruples of Fibonacci numbers
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.
(slides)
Isogénies entre jacobiennes de courbes hyperelliptiques
L'algorithme de la moyenne arithméticogéométrique, 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:
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 nonabelian
Chabauty method to \(p\)adic heights, enabling us to explicitly compute a
finite set of \(p\)adic points containing the rational points.
Torsion of elliptic curves over number fields
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.
(slides)
Sums of consecutive perfect powers is seldom a perfect power
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 \((x,y,n)\).
(slides)
Frey Curves, the Large Sieve and a Problem of Erdős
Consider the following Diophantine problem:
\[
n(n+d)(n+2d)\cdots (n+(k1)d)=y^\ell, \qquad \gcd(n,d)=1,
\]
where \(n\), \(d\), \(y\) are integers and the exponent \(\ell\) is prime. There are
obvious solutions with \(y=0\) or \(d=0\). A longstanding 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 Rothlike theorems on the existence
of \(3\)term arithmetic progressions in certain sets. This is joint work with
Mike Bennett.
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.
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 \(1\).
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.


 
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