## STNB2020(34th edition)

### Fermat's Last Theorem and Wendt's Resultant

#### Presenters

Alain Kraus

#### Abstract

Let $n$ be a positive integer and denote by $W_n$ the resultant of the polynomials $X^n-1$ and $(X+1)^n-1$.
At the end of the 19th century, Ernst Wendt noticed a connection between $W_n$ and Fermat's Last Theorem. In this talk, we will explore this relation over a totally real number field~$K$.
More precisely, for a prime number $p\geq 5$,
let $F_p/K$ be the Fermat curve defined by the equation $x^p+y^p+z^p=0$.
Using the modular method with Hilbert modular forms
over $K$ (as introduced by Siksek in the previous lecture),
we will discuss a criterion
involving $W_n$ that sometimes allows to prove, for concrete values of $p$, that $F_p(K)$ is trivial, i.e.
the only $K$-rational points of $F_p$ satisfy $xyz=0$.
This kind of question and criterion arise naturally as a complement
to the study of the
asymptotic Fermat's Conjecture presented by Siksek.
Finally, we will also discuss to what extent our criterion could conjecturally imply the asymptotic Fermat Conjecture over $K$.

#### Files

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