STNB2020(34th edition)

Fermat's Last Theorem and Wendt's Resultant


Alain Kraus


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 $K$.


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