The famous problem of determining all perfect powers in the Fibonacci sequence ${\left(F_n\right)}_{n\ge 0}$ and in the Lucas sequence ${\left(L_n\right)}_{n\ge 0}$ has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations $L_n=q^a{y}^p$, with $a\>0$ and $p\ge 2$, for all primes $q\<1087$ and indeed for all but $13$ primes $q\<{10}^6$. Here the strategy of [10] is not sufficient due to the sizes of...

We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves,...

We show that the Deligne formal model of the Drinfeld $p$-adic half-plane relative to a local field $F$ represents a moduli problem of polarized $O_F$-modules with an action of the ring of integers in a quadratic extension $E$ of $F$. The proof proceeds by establishing a comparison isomorphism with the Drinfeld moduli problem. This isomorphism reflects the accidental isomorphism of ${\mathrm{SL}}_2\left(F\right)$ and $\mathrm{SU}\left(C\right)\left(F\right)$ for a two-dimensional split hermitian space $C$ for $E/F$.

The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$

Let $E/Q$ be a modular elliptic curve, and let $K$ be an imaginary quadratic field. We show that the $p$-Selmer group of $E$ over certain finite anticyclotomic extensions of $K$, modulo the universal norms, is annihilated by the «characteristic ideal» of the universal norms modulo the Heegner points. We also extend this result to the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$. This refines in the current contest a result of [1].

In previous articles, we showed that for number fields in a certain large class, there are at most elliptic points on a Shimura curve of Γ₀(p)-type for every sufficiently large prime number p. In this article, we obtain an effective bound for such p.

Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside $S$.

We prove some new effective results of André-Oort type. In particular, we state certain uniform improvements of the main result in [L. Kühne, Ann. of Math. 176 (2012), 651-671]. We also show that the equation X + Y = 1 has no solution in singular moduli. As a by-product, we indicate a simple trick rendering André's proof of the André-Oort conjecture effective. A significantly new aspect is the usage of both the Siegel-Tatuzawa theorem and the weak effective lower bound on the class number of an...