The Genus of an Algebraic Curve

Edwards, Harold M.

doi:10.1007/978-3-030-98558-5_4

Harold M. Edwards³

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Abstract

Abel, lonely and unknown, was temporarily in Paris thanks to a travel grant from the government of Norway, and he hoped to win recognition in the city that was then the mathematical capital of Europe. Unfortunately, he naively believed that recognition could be won by submitting a work of undeniable genius to Europe’s leading mathematical institution. He did not understand that works of undeniable genius are inherently difficult to read, even for the most learned readers, and he did not understand that the members of Europe’s leading mathematical institution would not devote the needed time and thought to the work of a 24-year-old mathematician who was unknown to them and who came from a country they had scarcely heard of.

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Notes

1.
The last work Abel published was a brief note that contained a theorem from the memoir. Abel’s biographer Oystein Ore says that the theorem of that last brief note is the theorem of the memoir [72, p. 219], but it is far short of the theorem in the introduction of the memoir that I am discussing in this essay and that I take to be, in Ore’s phrase, “the main theorem from the Paris memoir.”
2.
This, in essence, is the theorem of Abel’s last published note that Ore mistook for the main theorem of the memoir. See the note above.
3.
In an effort to clarify Abel’s statement, I have taken some liberties with the translation. His actual words were, “Le nombre de ces relations ne dépend nullement du nombre des fonctions, mais seulement de la nature des fonctions particulière qu’on considère. Ainsi, par exemple, pour une fonction elliptique ce nombre est 1; pour une fonction dont la dérivée ne contient d’autres irrationalités qu’un radical du second degré, sous lequel la variable ne passe pas le cinquième ou sixième degré, le nombre des relations nécessaires est 2, et ainsi de suite.” His “fonctions” are the integrals above, and his “dérivées” are the integrands. What he is calling “une fonction elliptique” is what is today called an elliptic integral.
4.
One could also use the more restrictive, but perhaps more natural, definition in which the multipliers are required to be in ${\textbf {Z}}[x]$. Then an element would be integral over x in the sense defined above if and only if some integer multiple of it was integral in the more restrictive sense. Since $\frac{1}{2}$ is certainly a “function” without poles, the definition given above is the one that describes “rational functions without poles” on an algebraic curve.
5.
If $z_{1}^{n_{1}}$ can be expressed as a sum of multiples of lower powers of $z_{1}$ in which the multipliers are in ${\textbf {Q}}[x]$, and $z_{2}^{n_{2}}$ can be expressed as a sum of multiples of lower powers of $z_{2}$ in which the multipliers are in ${\textbf {Q}}[x]$, then every polynomial in $z_{1}$ and $z_{2}$ with coefficients in ${\textbf {Q}}[x]$ can be expressed as a sum of multiples of $z_{1}^{i}z_{2}^{j}$ with coefficients in ${\textbf {Q}}[x]$, where $i<n_{1}$ and $j<n_{2}$. Therefore, multiplication by any such polynomial in $z_{1}$ and $z_{2}$—in particular multiplication by $z_{1}+z_{2}$ and $z_{1}z_{2}$—can be represented by the $n_{1}n_{2}\times n_{1}n_{2}$ matrix of elements of ${\textbf {Q}}[x]$ that gives its effect on these $n_{1}n_{2}$ monomials $z_{1}^{i}z_{2}^{j}$. Therefore, since the polynomial in $z_{1}$ and $z_{2}$ is a root of the (monic) characteristic polynomial of this $n_{1}n_{2}\times n_{1}n_{2}$ matrix by the Cayley–Hamilton theorem, $z_{1}+z_{2}$ and $z_{1}z_{2}$, and, in the same way, all polynomials in $z_{1}$ and $z_{2}$ with coefficients in ${\textbf {Q}}[x]$, are integral over x.
6.
Note that $\chi (x, y)/x^{n\lambda }$ has the form $\chi _{1}(1/x, y/x^{\lambda })$, where $\chi _{1}$ is irreducible with integer coefficients and monic in its second variable, when $\lambda $ is large enough, and that x and y can be expressed rationally in terms of $u=1/x$ and $v=y/x^{\lambda }$, so the field of rational functions on $\chi (x, y)=0$ can also be regarded as the field of rational functions on the curve $\chi _{1}(u, v) =0$. To say that an element of this field is integral over 1/x means, of course, that it is integral over u.
7.
Newton’s presentation is quite sketchy. My main source was Walker [82]. See also Newton [71, vol. 3, p. 50 and p. 360, vol. 4, p. 629], Hensel–Landsberg [46], and Chebotarev [16]. Chebotarev cites (end of §2) the Hensel–Landsberg book as his basic source, but he examines the Newton polygon much more fully than that book does, dealing thoroughly, for instance, with the history of the method. Unfortunately, his article is available only in Russian. Chebotarev advocates calling it Newton’s “diagram” as Hensel and Landsberg do, saying that the “polygon” was not present in Newton’s formulation, but the name “Newton’s polygon” now seems firmly established.
8.
Walker’s proof of this point [82, p. 102] is not constructive, because he jumps from the observation that the ambiguity can never increase and can never go below 1 to the conclusion that he can find a step beyond which the ambiguity never decreases.
9.
This polynomial can be found because $\Psi (x, p)$ is a factor of the characteristic polynomial of the matrix that represents multiplication by p(x, y) relative to the basis $1, y, \ldots $, $y^{n-1}$.
10.
Strictly speaking, this construction does not apply to $\Psi (x,p)$, because its coefficients are rational and the description of the Newton polygon algorithm in Essay 4.4 assumes that the coefficients of the given equation $\chi (x, y)=0$ are integers, but the algorithm applies without modification to the case of rational coefficients. Moreover, $\chi (x, y)$ was assumed in Essay 4.4 to be irreducible. The series expansions of a reducible polynomial can be found by finding the expansions of its irreducible factors.
11.
What is to be shown is that the ratio $\sigma /\tau $ for any segment of the polygon is larger than the ratio $\sigma /\tau $ for the segment to its right. Since $\sigma /\tau $ is minus the slope of the segment, this is the statement that the slopes of the segments increase as one moves from left to right, which is evident. In actual inequalities, the three endpoints of two successive segments of Newton’s polygon, call them $(r, j_{r}), (s, j_{s}), (t, j_{t})$, satisfy
$$\begin{aligned}&\quad \sigma r+\tau j_{r}=\sigma s+\tau j_{s}<\sigma t+\tau j_{t},\\&\sigma 'r+\tau 'j_{r}>\sigma 's+\tau 'j_{s}=\sigma 't+\tau ^{\prime }j_{t}, \end{aligned}$$
where $\sigma '$ and $\tau '$ pertain to the segment from $(s, j_{s})$ to $(t, j_{t})$, from which
$$\begin{aligned} \tau (j_{r}-j_{s})=\sigma (s-r)\qquad \text{ and }\qquad \tau '(j_{r}-j_{s}) >\sigma '(s-r) \end{aligned}$$
follow. Therefore
$$\begin{aligned} \frac{\sigma }{\tau }=\frac{j_{r}-j_{s}}{s-r}>\frac{\sigma '}{\tau '}, \end{aligned}$$
as was to be shown.
12.
Multiplication of $\sum _{i=1}^{\mu }\frac{\phi _{i}(x)}{(x-a_{i})^{\nu _{i}}}=0$, where $a_{1}, a_{2}, \ldots , a_{\mu }$ are distinct algebraic numbers and $\deg \phi _{i}<\nu _{i}$ for each i, by $\prod _{i=1}^{\mu }(x-a_{i})^{\nu _{i}}$ gives an equation $\sum _{i=1}^{\mu }\psi _{i}(x)=0$ in which the $\psi _{i}(x)$ are polynomials. All but one of these polynomials is divisible by $(x-a_{1})^{\nu _{1}}$, so the remaining one must also be divisible by $(x-a_{1})^{\nu _1}$, from which it follows that $\phi _{1}(x)$ must be zero. In the same way, $\phi _{i}(x)=0$ for each i.
13.
Such an embedding of an algebraic field of transcendence degree 1 in ${\textbf {A}}\langle s\rangle $ is analogous to an embedding of an algebraic number field in the field of complex numbers (see Essay 5.1). As in the latter case, the field $\hat{K}$ loses much of its constructive meaning when it is regarded merely as a subfield of ${\textbf {A}}\langle s\rangle $, because elements of ${\textbf {A}}\langle s\rangle $ are infinite series. An element of $\hat{K}$ is a root of a polynomial whose coefficients are rational functions of x, so the infinite series that represents it as an element of ${\textbf {A}}\langle s\rangle $ can be specified by giving enough terms to determine an unambiguous truncated solution of the equation in question, after which all later terms are determined by Newton’s polygon.
14.
As always, ${\textbf {Q}}(x)$ denotes the field of quotients of ${\textbf {Z}}[x]$, which is to say the field of rational functions in x.
15.
It is natural to exclude $\delta _{i}=0$, because the points $(x_{i}, y_{i})$ for which $\delta _{i}=0$ can be omitted from the list.
16.
See Essay 2.2. The field ${\textbf {Q}}(z)$ of rational functions in z is isomorphic to a subfield of the root field, and the root field has finite degree over the subfield provided z is not a constant. Adjunction of x and y to ${\textbf {Q}}(z)$ gives an explicit extension of ${\textbf {Q}}(z)$ that is isomorphic to the root field of $\chi (x, y)$. By the theorem of the primitive element, such a double adjunction can be obtained by a simple adjunction, and one can describe a simple adjunction as the root field of an irreducible monic polynomial with coefficients in ${\textbf {Z}}[z]$ in the usual way.
17.
Since the root field is an extension of ${\textbf {Q}}(x)$ of finite degree n, the powers 1, $z, z^{2}, \ldots , z^{n}$ are linearly dependent over ${\textbf {Q}}(x)$, which is to say that one can find a nonzero polynomial of degree at most n with coefficients in ${\textbf {Q}}(x)$ of which z is a root. The needed relation $\phi (x, z)=0$ is found by clearing denominators and passing to an irreducible factor if necessary. Because the root field of $\chi (x, y)$ contains ${\textbf {Q}}(x)$, the relation $\phi (x, z)=0$ must involve z. To say that z is a parameter—that it is not a constant—means that $\phi $ also involves x; if this is not the case, the denominator of $\frac{dx}{dz}$ is zero and the derivative is not defined (x is not a function of z).
18.
This definition is imprecise in that it ignores the question of multiplicities. In Essay 4.7, $n=\deg _{y}\chi $ distinct embeddings of the root field of $\chi $ in ${\textbf {A}}\langle s\rangle $ were constructed for each given $\alpha $. A given f may have fewer than n distinct images, so the same principal parts may occur for more than one embedding. Obviously, Lemma 1 is not affected by the way in which multiplicities are treated.

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Harold M. Edwards

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Edwards, H.M. (2022). The Genus of an Algebraic Curve. In: Essays in Constructive Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-98558-5_4

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DOI: https://doi.org/10.1007/978-3-030-98558-5_4
Published: 29 September 2022
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