Why is it "easier" to work with function fields than with algebraic number fields?

Question

I just bought a copy of Jürgen Neukirch's book Algebraic Number Theory. While browsing through it I found a section titled § 14. Function Fields in chapter I. In it the author describes some aspects of an analogy between function fields and algebraic number fields.

This led me to google for a while and I ended up reading the Wikipedia entry for Global Field. And this is where my question comes from. In the last sentence of that entry there's the following passage, which I find really interesting:

It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example.

Unfortunately, being as dramatic as it is, the example mentioned does not tell me anything because not even the Wikipedia entry on Arakelov Theory is somehow close to give even a small hint as to what it is about.

So I would like to ask for some insight and/or examples that illustrate why it is said to be easier to work with function fields than with algebraic number fields and then try to develop parallel techniques for the number field case.

Thank you very much for any help.

Answer 1

One answer is that we can take formal derivatives. For example, Fermat's last theorem is rather difficult but the function field version is a straightforward consequence of the Mason-Stothers theorem, whose elementary proof crucially relies on the ability to take formal derivatives of polynomials.

There is no obvious way to extend this construction to integers in a way that preserves its good properties. If there were, then the abc conjecture (of which Mason-Stothers is the function field version) would be trivial, which it's not. There is a thing called the arithmetic derivative, but it is of course not linear, and it doesn't seem to me to be very easy to prove anything with it.

The problem is that if we want to think of $\mathbb{Z}$ as being analogous to a function field, then the "field" that it's a function field over is the field with one element, so if a reasonable notion of formal derivative exists here it needs not to be $\mathbb{Z}$-linear, but to be $\mathbb{F}_1$-linear, whatever that means... if we understood what that meant, perhaps we could construct the "correct" version of the arithmetic derivative and presumably prove the abc conjecture.

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Arakelov theory addresses another difference between function fields and number fields, which is the existence of Archimedean places. Over a function field all places are non-Archimedean and I understand this makes various things easier, but I don't know much about this so someone else should chime in here.

Answer 2

A primary reason that function field arithmetic is simpler than number field arithmetic is due to the existence of nontrivial derivations. With the availability of derivatives many things simplify.

E.g. for polynomials derivatives yield easy algorithms for squarefree testing, squarefree part, etc. Contrast this to the integer case: no feasible (polynomial time) algorithm is currently known for recognizing squarefree integers or for computing the squarefree part of an integer. In fact it may be the case that this problem is no easier than the general problem of integer factorization. This problem is important because one of the main tasks of computational algebraic number theory reduces to it (in deterministic polynomial time). Namely the problem of computing the ring of integers of an algebraic number field depends upon the square-free decomposition of the polynomial discriminant when computing an integral basis.

From derivatives also come Wronskians and associated measures of independence (excerpted below). For example, this is what is at the heart of Mason's trivial high-school level proof of the ABC theorem for polynomials - which is a difficult important open problem for numbers. From Mason's theorem follows immediately a trivial two-line proof of FLT for polynomials (excerpted below). If there existed some sort of analogous "derivative for integers" that yielded the corresponding ABC theorem, then it would yield an analogous trivial proof of FLT for integers (more precisely it would yield asymptotic FLT, i.e. FLT for all sufficiently large exponents).

Such observations have motivated searches for "arithmetic analogues of derivations". For example, see Buium's paper by that name in Jnl. Algebra, 198, 1997, 290-99, and see his book Arithmetic differential equations.

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Subject: poly FLT, abc theorem, Wronskian formalism [was: Entire solutions of f^2+g^2=1]
Author: "Bill Dubuque" <[email protected]>
Posted to sci.math on Jul 17, 1996 12:13 AM

"Harold P. Boas" [email protected] wrote to sci.math.research on 7/3/96:

Robert Israel wrote:

Alan Horwitz [email protected] wrote:

I am interested in all entire solutions $f$ and $g$ to $f^2+g^2=1$.
I remember seeing this somewhere, but I cannot recall where.

I've also seen this before, in fact I recall assigning it as homework to one of my classes, but I don't recall the source. The solutions are

Robert B. Burckel gives some history about this problem in his
comprehensive book An Introduction to Classical Complex Analysis,
volume 1 (Academic Press, 1979). In Theorem 12.20, pages 433-435,
he shows that the equation $f^n+g^n=1$ has no nonconstant entire
solutions when the integer $n$ exceeds $2$; when $n=2$, the solution
is as given by R. Israel in his post. ... (papers of Fred Gross)

Note that the rational function case of FLT follows trivially from Mason's abc theorem, e.g. see Lang's Algebra, 3rd Ed. p. 195 for a short elementary (high-school level) proof of both. Chebyshev also gave a proof of FLT for poly's via the theory of integration in finite terms, e.g. see p. 145 of Dane Shanks' Solved and Unsolved Problems in Number Theory, or Ritt's Integration in Finite Terms, p. 37. The Chebyshev result is actually employed as a subroutine of Macsyma's integration algorithm (implemented decades ago by Joel Moses). Via $abc$ a related result of Dwork is also easily proved: if $A,B,C$ are fixed poly's then all coprime poly solutions of $A X^a+B Y^b+C Z^c = 0$ have bounded degrees provided $1/a+1/b+1/c < 1$. Other applications in both number and function fields may be found in Lang's survey [3].

Mason's $abc$ theorem may be viewed as a very special instance of a Wronskian estimate: in Lang's proof the corresponding Wronskian identity is $\,c^3 W(a,b,c) = W(W(a,c),W(b,c)),\,$ thus if $a,b,c$ are linearly dependent then so are $W(a,c),W(b,c).\,$ The sought bounds follow upon multiplying the latter dependence relation through by $\,N_0 = r(a) r(b) r(c),\,$ where $\,r(x) = x/\gcd(x,x')$.

More powerful Wronskian estimates with applications toward diophantine approximation of solutions of LDEs may be found in the work of the Chudnovsky's 1 and C. Osgood 2. References to recent work may be found (as usual) by following MR citations to these papers in the MathSci database.

I have not seen mention of this Wronskian view of Mason's abc theorem. Although elementary, it deserves attention since it connects the abc theorem with the general unified viewpoint of the Wronskian formalism as proposed by the Chudnovsky's and others.

1 Chudnovsky, D. V.; Chudnovsky, G. V. The Wronskian formalism for linear differential equations and Pade approximations. Adv. in Math. 53 (1984), no. 1, 28--54. 86i:11038 11J91 11J99 34A30 41A21

2 Osgood, Charles F. Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better. J. Number Theory 21 (1985), no. 3, 347--389. 87f:11046 11J61 12H05

[3] Lang, Serge Old and new conjectured Diophantine inequalities. Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 1, 37--75. 90k:11032 11D75 11-02 11D72 11J25

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Bill Dubuque wrote on sci.math on Sep 25, 1998, 3:00:00 AM
$\ \ $ John Scholes wrote: (I've generalized the expt from 10 to $n > 1\ $ -Bill)

Find all real $p,q,a,b$ so $(2x-1)^2n - (ax+b)^2n = (x^2+px+q)^n$ for all $x$.

By Mason's abc theorem (1), if the polys share no root, their degrees are smaller than the total count of all their distinct roots, so $2n < 1+1+2$. But $n \ge 2$ so the polys must all share the root $x=1/2$. The rest is easy.

Mason's abc theorem is very powerful, e.g. see below the one-line proof of the polynomial version of Fermat's Last Theorem (FLT).

(1) Mason's abc Theorem (1984). Let $a(x), b(x), c(x)$ in $\Bbb C[x]$ be coprime polynomials with $a + b = c$. Then

$$\max \deg\{a,b,c\} \le N(abc)-1,\,\ N(p): = \text{number of distinct roots of $p$ in $\Bbb C$}$$

(2) Corollary (summing the above over $a,b,c$)

$$\deg(abc) \le 3 (N(abc)-1)$$

This elementary result yields much power: witness this trivial proof of

FLT for polynomials: if $p^n + q^n = r^n$ with $p,q,r$ coprime, by (2)

$$n \deg(pqr) = deg(abc) \le 3 N(abc) - 3 \le 3 \deg(pqr) - 3$$

therefore $\ (3-n) \deg(pqr) \ge 3\,$ so $\,n < 3$.

The proof of (1) is easy, requiring only a half-page of high-school algebra, see, for example: Lang, Serge. Algebra. 3rd Edition. 1993. S.IV.7 p.194 (or see the above Wronskian form).

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Below is further info which may be of interest - both mathematically and historically. It is an excerpt of an email I sent to Franz Lemmermeyer after seeing mention of Snyder's similar proof (2000) of Mason's $\rm ABC$ theorem in some of Franz's lecture notes. Later I posted this email to sci.math post on un 26, 2010 while lamenting the frequent failure of authors to cite (obvious) electronic sources.

[Below is an excerpt of an email I sent to Franz Lemmermeyer on 14 Mar 2005, slightly edited]

Hi, I just noticed a reference to Noah Snyder's proof of Mason's ABC theorem in your lecture. I wonder if this is really any different than the Wronskian viewpoint that I've pointed out since the mid 80's, which I have mentioned in passing in various places online since at least '96 e.g. sci.math, and one of many (Wolfram) MathWorld pages based wholly on some of my posts to (Schroeppel's) math-fun mailing list (some, alas, uncredited). Do you know how I might obtain a copy of Snyder's article? [later] Thanks so much for making Synder's paper available. As I suspected Snyder's proof is essentially the same proof I gave over 20 years ago. I've mentioned this proof online in many places, e.g. in 1996 on math-fun & sci.math

[edit: snipped above sketched proof from my sci.math post]

This is precisely what Snyder does in his proof. The first hit on Googling "Mason's theorem" is said MathWorld page which excerpts my post on math-fun, and gives a url link to my 1996 sci.math post sketching the proof. Thus I'm surprised that Snyder and his mentors/editors didn't know this.

There are actually much deeper things one can do from this Wronskian viewpoint - it is a fundamental approximation tool in differential algebra. In the mid eighties I was working for the Macsyma group on effective approaches for special functions using tools from differential algebra, so I was quite familiar with Wronskian tools. Hence it was a nice confluence of events that Mason's work happened around the same time, since I immediately recognized the relationship. If I'm lucky enough to re-obtain my math library I should dig up some of my notes on this and write a letter to the editor since there are still some things worthy of mention.

It seems that Franz agrees with my assessment, since he later wrote in online lecture notes

A few years ago, the high school student (now Harvard undergraduate) Noah Snyder [An alternate proof of Mason’s theorem, Elem. Math. 55 (2000), 93–94] came up with a ‘proof from the Book’ for Mason’s ABC theorem (actually it is very close to a proof posted by Bill Dubuque to sci.math a few years earlier).

It is unfortunate that I hadn't seen Noah's paper before publication because if so I probably could have nudged him to include mention of the Wronskian viewpoint - a powerful tool that deserves to be much better known. Hopefully the above will help alleviate that.

Answer 3

Let's consider an example to see why function fields are easier:

Let $q$ be a prime, and consider the global function field, $\mathbb F_q(T)$. An ideal $\mathfrak a$ of $\mathbb F_q[T]$ is just the principal ideal $\mathfrak a=(f)=(T^d+a_{d-1}T^{d-1}+\dots+a_0)$. The norm $N\mathfrak a=q^d$, and you can see that there are exactly $q^d$ ideals of norm $q^d$.

Then the zeta function over this field is $$\zeta_{\mathbb F_q[T]}(s)=\sum_{\mathfrak a \neq 0}N\mathfrak a^{-s}=\sum_{d=0}^\infty q^d(q^d)^{-s}=1/(1-q^{1-s})$$

That's a very simple expression for the zeta function. Note that it has no zeros, so it trivially satisfies the Riemann Hypothesis.

Answer 4

In number theory one is interested in counting different objects which have arithmetic significance. When working over function fields, this problem can usually be translated into counting points of an algebraic variety over a finite field. One then has a host of techniques available to study such numbers, most importantly, l-adic cohomology. For example, orbital integrals are about counting certain arithmetically significant finite set. Over global fields, this problems translates to point-counts on variants of affine Springer fibres. This is the starting point for Ngo's proof of the Fundamental Lemma.

Answer 5

Here is one aspect of the difference in Arakelov theory: In the number field case we have a "naive" Riemann-Roch formula: $$ \chi(\alpha)=-\log \textrm{Vol}(\alpha) $$ where $\alpha$ is an Arakelov divisor on $K$. Using the explicit formula for $\textrm{Vol}(\alpha)$ we can re-write this as $$ \chi(\alpha)-\chi(O_{K})=\deg(\alpha) $$ And using Poisson summation formula, we can define $h^{0}(\alpha), h^{1}(\alpha)$ such that $$ h^{0}(\alpha)=\log(\sum_{f\in I}e^{-\pi|f|_{\alpha}^{2}}), h^{1}(\alpha)=h^{0}(K-\alpha), h^{0}-h^{1}=\deg(\alpha)-\frac{1}{2}|\Delta| $$ But the corresponding construction in the functional field case is radically different. As we know from Hodge theorem: $$ H^{1}(X, L)\cong \ker \Delta^{0,1}(L) $$ and in particular its number is an non-negative integer. Formally we do have an analgous McKean-Singer formula: $$ \chi(X,L)=Tr(e^{-D^{*}D})-Tr(e^{-DD^{*}}) $$ But the analogy is not perfect: While both formulas are related to the trace of heat kernel coming from a Dirac operator $D$, there seems to be no good way for us to interpret $h^{0}(\alpha)$ as the dimension of the kernel of some linear operator because of the presence of the $\log$ term in the front. Similarly other approaches of bridiging the Riemann-Roch in number field case and functional field case collapses: There is no good analog of derived functor cohomology that defines $H^{0}, H^{1}$ in the number field case. The best analogy we have for Cech cohomology is theory of ghost spaces, and it is not even a group. On the other hand, translating ghost space machinery to function field case is also not easy. Thus the correspondence of most common cohomology theories (de Rham, Cech, derived functor, singular/betti) between the two worlds breaks down quite badly. The functional field case is "infinitely better" because of certain level of smoothness than the number field.

Rather, the analogy comes from elsewhere: Recall that we have the Riemann singularity theorem:

Let $C$ be a smooth curve of genus $g$, then for every effective divisor of degree $g-1$: $$ \textrm{multi}_{\mu(D)+K}(\Theta)=h^{0}(C,O(D)) $$

Van Der Geer and Rene Schoof viewed this as the their motivation for the Riemann-Roch in the number field case (see page 7, for example). Personally I think what is really remarkable is that the analogy persists even in arithemetic surface case, that we have the Faltings volume on the determinant of cohomology defined by the pull-back of the metric of the line bundle $O_{J}(-\Theta)$. While there is no higher dimensional analog of this fact in Arakelov theory (we have to replace Faltings volume by Quillen metric using analytic torsion), this nevertheless suggests a very deep and sutble role played by $\theta$ functions both in the number field and functional field case.