English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In 2000, Darmon described a program to study the generalized Fermat equationusing modularity of abelian varieties of $\mathrm{GL}_2$-type over totally realfields. The original approach was based on hard open conjectures, which havemade it difficult to apply in practice. In this paper, building on the progresssurrounding the modular method from the last two decades, we analyze and expandthe current limits of this program by developing all the necessary ingredientsto use Frey abelian varieties for new Diophantine applications. In particular,we deal with all but the fifth and last step in the modular method for Fermatequations of signature $(r,r,p)$ in almost full generality.  As an application, for all integers $n \geq 2$, we give a resolution of thegeneralized Fermat equation $x^{11} + y^{11} = z^n$ for solutions $(a,b,c)$such that $a + b$ satisfies certain $2$- or $11$-adic conditions.  Moreover, the tools developed can be viewed as an advance in addressing adifficulty not treated in Darmon's original program: even assuming `big image'conjectures about residual Galois representations, one still needs to find amethod to eliminate Hilbert newforms at the Serre level which do not havecomplex multiplication. In fact, we are able to reduce the problem of solving$x^5 + y^5 = z^p$ to Darmon's `big image conjecture', thus completing a line ofideas suggested in his original program, and notably only needing the Cartancase of his conjecture.

On Darmon's program for the generalized Fermat equation, I