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I asked this question on MSE but didn't get any response, so I'm asking here. I apologize in advance if this question is not research level.

A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two different elliptic curves. To transform the Fermat Curve of degree $3$, the substitutions $$ a=\frac{12z}{x+y},\quad b=\frac{36(x-y)}{x+y} $$ produce $b^2=a^3-432$, an elliptic curve. Similarly for the Fermat Curve of degree $4$, the substitutions $$ a=\frac{2(y^2+z^2)}{x^2},\quad b=\frac{4y(y^2+z^2)}{x^3} $$ give $b^2=a^3-4a$. However, the substitutions used are not at all obvious, which leads me to wonder,

Is there a similar substitution which can relate a Fermat curve of arbitrary degree to an elliptic curve?

This is equivalent to asking whether there is always a nonconstant morphism from a Fermat curve to an elliptic curve.

Thank you in advance!

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    $\begingroup$ @AdamP.Goucher The OP is asking for a morphism from the Fermat curve onto an elliptic curve, not an isomorphism. Equivalently, does the Jacobian of the Fermat curve have an elliptic factor? $\endgroup$
    – Joe Silverman
    Commented Dec 20, 2018 at 2:54
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    $\begingroup$ The cubic Fermat curve is a smooth cubic in $\mathbb P^2$, so has genus 1. It has the rational point $[1,-1,0]$. Hence it is isomorphic to an elliptic curve given by a Weierstrass equation. Finding the transformation is standard. The quartic Fermat curve maps 2-to-1 to the curve $C:u^4+1=v^2$ via $u=x/y$ and $v=z^2/y$. The curve $C$ also has genus 1 and a rational point $(u,v)=(0,1)$, hence it too can be mapped to an elliptic curve in Weierstrass form (using 19th century formulas!) . For higher $n$, there are lower genus curves that the Fermat curve maps to, but they're generally not elliptic. $\endgroup$
    – Joe Silverman
    Commented Dec 20, 2018 at 3:00
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    $\begingroup$ The Jacobian of the Fermat curve decomposes (up to isogeny) as a product of CM abelian varieties. Accordingly the motive of the Fermat curve $X_N$ decomposes as a direct sum of motives associated to Hecke characters of the cyclotomic field $\mathbf{Q}(\zeta_N)$ (see Otsubo's work e.g. On special values of Jacobi-sum Hecke $L$-functions). So your question is about rationality properties of these Hecke characters. $\endgroup$
    – François Brunault
    Commented Dec 20, 2018 at 9:24
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    $\begingroup$ I don't know the answer to the question but here is a reference: Koblitz, Rohrlich, Simple factors in the Jacobian of a Fermat curve. If I understand Theorem 2 correctly, then if $N \geq 5$ is a prime $\equiv 2 \textrm{ mod } 3$ then the simple factors of the Jacobian of $X_N$ all have dimension $(N-1)/2$. So for exemple $X_5$ and $X_{11}$ do not map to any elliptic curve. $\endgroup$
    – François Brunault
    Commented Dec 20, 2018 at 9:38
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    $\begingroup$ @FrançoisBrunault: I think you should write your comment as an answer, this is actually quite nontrivial. The paper you quote is freely available at https://cms.math.ca/openaccess/cjm/v30/cjm1978v30.1183-1205.pdf. $\endgroup$
    – abx
    Commented Dec 20, 2018 at 10:08

1 Answer 1

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The following reference seems to answer the question of which Fermat curves admit a non-constant map to an elliptic curve:

Neal Koblitz, David Rohrlich, Simple factors in the Jacobian of a Fermat curve, Canadian Journal of Mathematics 30 No. 6 (1978) pp. 1183–1205, doi:10.4153/CJM-1978-099-6 (free pdf).

As a particular case of Theorem 2 there, if $N \geq 5$ is a prime $\equiv 2 \textrm{ mod } 3$ then the simple factors of the Jacobian $J_N$ of the Fermat curve $X_N$ all have dimension $(N−1)/2$. So for example, the Fermat curves $X_5$ and $X_{11}$ do not map to any elliptic curve.

The motive of the Fermat curve $X_N$ has been extensively studied: it decomposes as a direct sum of motives associated to Hecke characters of the cyclotomic field $\mathbf{Q}(\zeta_N)$, see e.g. Otsubo's work On special values of Jacobi-sum Hecke $L$-functions. Note also that every elliptic factor of a Fermat curve must have complex multiplication, essentially because $X_N$, and thus its Jacobian $J_N$, admits an action of the rather large group $\mu_N \times \mu_N$, where $\mu_N$ is the group of $N$-th roots of unity in $\mathbf{C}$. More generally, the factors of $J_N$ are CM abelian varieties.

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