The generalized Fermat equation with exponents 2, 3, $n$

@article{Freitas2019TheGF,
  title={The generalized Fermat equation with exponents 2, 3, \$n\$},
  author={Nuno Freitas and Bartosz Naskręcki and Michael Stoll},
  journal={Compositio Mathematica},
  year={2019},
  volume={156},
  pages={77 - 113}
}
We study the generalized Fermat equation $x^{2}+y^{3}=z^{p}$ , to be solved in coprime integers, where $p\geqslant 7$ is prime. Modularity and level-lowering techniques reduce the problem to the determination of the sets of rational points satisfying certain 2-adic and 3-adic conditions on a finite set of twists of the modular curve $X(p)$ . We develop new local criteria to decide if two elliptic curves with certain types of potentially good reduction at 2 and 3 can have symplectically or anti… 

Sums of two cubes as twisted perfect powers, revisited

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^\alpha C^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for

On symplectic isomorphims of the p-torsion of elliptic curves

Let $\ell$ and $p$ be distinct prime numbers with $p\geq 3$. Let $E/\mathbb{Q}_{\ell}$ and $E'/\mathbb{Q}_{\ell}$ be elliptic curves defined over $\mathbb{Q}_{\ell}$, having potentially good

On a class of generalized Fermat equations of signature (2,2n,3)

Cartan modular curves of level 13

We give explicit equations for the modular curves $X_\text{s}(13)$ and $X_\text{ns}(13)$ associated respectively to a split and a non-split Cartan subgroup of $\text{GL}_2(\mathbb F_{13})$. We

AN APPLICATION OF THE MODULAR METHOD AND THE SYMPLECTIC ARGUMENT TO A LEBESGUE–NAGELL EQUATION

In this paper, we study the generalized Lebesgue-Nagell equation \[ x^2+7^{2k+1}=y^n. \] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $k\geq0$ and $q>0$ where

Chabauty Without the Mordell-Weil Group

Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve C, given only the p-Selmer group S of its

La corba de Frey: teoria i aplicacions

TLDR
This thesis starts with a brief study on the Rieman-Roch Theorem, then develops the construction of Frey’s curve and study some of its properties, and gives a short introduction to modular functions and Galois representation.

On Galois inertial types of elliptic curves over $\mathbb{Q}_\ell$

We provide a complete, explicit description of the inertial Weil–Deligne types arising from elliptic curves over Ql for l prime.

On the symplectic type of isomorphisms of the 𝑝-torsion of elliptic curves

TLDR
Let P greater-than-or-equal-to 3 and E prime slash double-struck upper Q be elliptic curves with isomorphic <inline-formula content-type="math/mathml" xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext) be a prime.

Double covers of Cartan modular curves

References

SHOWING 1-10 OF 104 REFERENCES

On the Fermat-type Equation $x^3 + y^3 = z^p$

We prove that the Fermat-type equation $x^3 + y^3 = z^p$ has no solutions $(a,b,c)$ satisfying $abc \ne 0$ and $\gcd(a,b,c)=1$ when $-3$ is not a square mod~$p$. This improves to approximately

Sums of two cubes as twisted perfect powers, revisited

In this paper, we sharpen earlier work of the first author, Luca and Mulholland, showing that the Diophantine equation $$ A^3+B^3 = q^\alpha C^p, \, \, ABC \neq 0, \, \, \gcd (A,B) =1, $$ has, for

Fermat's Last Theorem over some small real quadratic fields

Using modularity, level lowering, and explicit computations with Hilbert modular forms, Galois representations and ray class groups, we show that for $3 \le d \le 23$ squarefree, $d \ne 5$, $17$, the

Classical and modular methods applied to Diophantine equations

TLDR
This work constructs certain families of Frey curves and proves irreducibility results for the Galois representation associated to the p-torsion points of some of these curves for small primes p and constructs an algorithm to solve the generalized Fermat equations with signature (2,3,5) and arbitrary nonzero integer coefficients.

Explicit Chabauty—Kim for the split Cartan modular curve of level 13

We extend the explicit quadratic Chabauty methods developed in previous work by the first two authors to the case of non-hyperelliptic curves. This results in an algorithm to compute the rational

Twists of X(7) and primitive solutions to x^2+y^3=z^7

We find the primitive integer solutions to x^2+y^3=z^7. A nonabelian descent argument involving the simple group of order 168 reduces the problem to the determination of the set of rational points on

Rational points on X_0^+ (p^r)

We show how the recent isogeny bounds due to E. Gaudron and G. Remond allow to obtain the triviality of X_0^+ (p^r)(Q), for r>1 and p a prime exceeding 2.10^{11}. This includes the case of the curves

Rational isogenies of prime degree

In this table, g is the genus of Xo(N), and v the number of noncuspidal rational points of Xo(N) (which is, in effect, the number of rational N-isogenies classified up to "twist"). For an excellent
...