Fermat-Goss-Denis theorem

Fermat's last theorem is the claim that $x ^ { n } - y ^ { n } = z ^ { n }$ has no solutions in non-zero integers for $n > 2$ (see also Fermat last theorem). However, over a function field $\mathbf{F} ( T )$ (cf. also Algebraic function), with $\mathbf{F}$ of non-zero characteristic $p$, the appropriate generalization is not just to take $x$, $y$ and $z$ as polynomials over $\mathbf{F}$ in $T$. In any event, in characteristic zero, or for $n$ prime to the characteristic $p$, it is fairly easy to see, by descent on the degrees of a putative solution $( x ( T ) , y ( T ) , z ( T ) )$, that there is not even a non-trivial solution over $\mathbf{F} [ T ]$, with $\mathbf{F}$ the algebraic closure of $\mathbf{F}$.

In 1982, D. Goss [a1] formulated a suitable analogue for the case $\operatorname { gcd } ( n , p ) \neq 1$. Goss notes that, traditionally, Fermat's equation is viewed as $y ^ { n } ( ( x / y ) ^ { n } - 1 ) = z ^ { n }$, where the connection with cyclotomic fields, and thence the classical exponential function, is displayed: the zeros of $w ^ { n } - 1$ are precisely the $n ^ { \text { th } }$ roots of unity. But in characteristic $p > 0$ the analogue of the exponential function comes by way of the Drinfel'd module; more specifically, the Carlitz module. A familiar and elementary manifestation of such things is the Hilbert theorem 90, whereby a cyclic extension of degree $p$ in characteristic $p$ is not given by a zero of $X ^ { p } - a$, but of $X ^ { p } - X - a$.

Let $\mathbf{F} = \mathbf{F} _ { q }$ be the field of $q = p ^ { m }$ elements. The equation that in this context appears to raise issues analogous to those provoked by the classical Fermat equation is

\begin{equation*} y ^ { q ^ { r } } \phi_f ( x / y ) - z ^ { p } = 0, \end{equation*}

where $\phi$ is the Carlitz module determined by $\phi _ { T } = T F ^ { 0 } + F$ and $F$ denotes the Frobenius mapping relative to $\mathbf{F} _ { q }$, i.e. the mapping that gives $q ^ { \text{th} }$ powers. To say that $\phi$ is the Carlitz module is to require also that $\phi _ { f } \phi _ { g } = \phi _ { f g }$. Goss [a1] deals, à la Kummer, with the case of this equation when $f$ is a regular prime of $\mathbf{F} _ { q } [ T ]$.

The equation has two important parameters, the element $f$ of $\mathbf{F} _ { q } [ T ]$ and the order $q = p ^ { m }$. As usual, a solution with $x y z \neq 0$ is called non-trivial. When $\operatorname { deg } f = 1$, Goss shows that in analogy with the equation $x ^ { 2 } - y ^ { 2 } = z ^ { 2 }$ there are an infinity of solutions. Suppose $f$ is monic. L. Denis [a2] proves that if $q \geq 3$, $p \neq 2$ and $\operatorname { deg } f \geq 2$, there is no non-trivial solution. If $q \geq 4$, $p = 2$ and $\operatorname { deg } f \geq 2$, there is a unique solution proportional in $\mathbf{F} _ { q }$ to the triplet $( 1,1 , T + T ^ { q / 2 } )$ in the case $f = T ^ { 2 } + T + \beta$, where $\beta$ is a square in $\mathbf{F} _ { q }$; and if $q = 2$, $\operatorname { deg } f \geq 4$, then there is a solution only if $f$ is of the shape $( T ^ { 2 } + T ) g ( T ) + 1$, and it is $( 1,1,1 )$. Denis deals completely with the remaining cases $q = 2$. Because $\phi$ is $\mathbf{F} _ { q }$-linear, one can now easily produce the results for $f$ not monic.

In settling the general case, Denis [a2] speaks of the Fermat–Goss theorem. It seems appropriate here to write of the Fermat–Goss–Denis theorem.

References

[a1]	D. Goss, "On a Fermat equation arising in the arithmetic theory of function fields" Math. Ann. , 261 (1982) pp. 269–286
[a2]	L. Denis, "Le théorème de Fermat–Goss" Trans. Amer. Math. Soc. , 343 (1994) pp. 713–726