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In number theory there is often an analogue between statements which holds over a number field (that is, a finite field extension $K/\mathbb{Q}$) and function fields (that is, finite extensions of the form $K/\mathbb{F}_q(t)$ where $q = p^k$ for some prime $p$ and $k \geq 1$). One of the most famous examples of such an analogue is the Riemann hypothesis. Weil showed that the Riemann Hypothesis holds for the function field case. However, progress on the function field case has not shed much, if any, light on the corresponding case for number fields.

An example of where the analogy breaks down (conjecturally) is the Brauer-Manin obstruction. For hypersurfaces defined over $\mathbb{P}^n(K)$ for some number field $K$, it is expected that most hypersurfaces $X$ of degree $d \geq n+1$ will be of general type, and hence neither satisfy the Hasse principle nor have its failure to satisfy the Hasse principle accounted for by the Brauer-Manin obstruction. Harari and Voloch showed that for function fields of positive characteristic, this is not the case: indeed in this paper (http://www.ma.utexas.edu/users/voloch/Preprints/carpobs3.pdf) they showed that the Brauer-Manin obstruction is the only reason Hasse principle can fail in this case.

Another case where the analogy is not convincing is ranks of elliptic curves. It is known that elliptic curves over a function field of positive characteristic may have arbitrarily large rank. However, this question is not known even conjecturally in the number field case. Indeed it seems that many experts disagree on this question. At a recent summer school on counting arithmetic objects in Montreal, Bjorn Poonen gave a take on a heuristic suggesting that elliptic curves over number fields should have bounded rank. Andrew Granville, one of the organizers, agrees with this assertion. However, other experts present including Manjul Bhargava disagreed.

My question is, what are some other situations where one expects genuinely different behavior between the function field setting and the number field setting?

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    $\begingroup$ Whilst I agree that the result of Harari and Voloch illustrates a nice difference between what is known for number fields and function fields; your interpretation of this result given by your application to hypersurfaces is not correct. They emphasise at numerous points in their paper that the results only apply to integral points on affine varieties, in particular their results do not apply in the case which you mention. $\endgroup$ – Daniel Loughran Jul 28 '14 at 11:45
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    $\begingroup$ Diophantine approximation. Thue-Siegel-Roth fails and there are many possibilities for the approximation exponent. $\endgroup$ – Felipe Voloch Jul 28 '14 at 13:23
  • $\begingroup$ The comments to my answer at mathoverflow.net/questions/3269/… give an example where a result about sums of squares in $\mathbf Z$ is limited to three squares but goes through in the case of $\mathbf F_q[u]$ to any finite sum of squares, essentially because the single unbounded place on $\mathbf F_q[u]$ is non-archimedean, whereas in the case of $\mathbf Z$ it's archimedean. $\endgroup$ – KConrad Jul 29 '14 at 12:34
  • $\begingroup$ Reading Weil's "Basic Number theory" I always feel that this book would have improved if Weil didn't stretch this analogy as far as he did. $\endgroup$ – Alex Gavrilov Aug 16 '18 at 7:26
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Instead of the final results, let me focus on the underlying reasons why number fields and function fields are different.

A. Every function field has subfields of arbitrarily large index (e.g. by taking the field generated by a rational function of large degree). But each number field has a subfield of maximal (finite) index.

This actually explains the discrepancy in the bounded ranks heuristic. The heuristic, being probabilistic, is not expected to apply to a special family of curves that has unusually high rank for a good reason. For instance, if you take an extension of $\mathbb Q$ with Galois group $(\mathbb Z/2)^n$, by looking at root numbers you can see that the average rank of curves in the family is at least $2^{n-1}$. The constructions of curves of large rank over function fields all, I believe, involve a similar pullback - but the pullback is from a subfield of $\mathbb F_q(t)$ to $\mathbb F_q(t)$.

B. There exist isotrivial objects over function fields. These have properties that cannot occur over number fields, because the fact that each prime number is different places a lower bound on how similar the reductions of a variety modulo different places can be.

For instance, there are no elliptic curves with good reduction at every place of $\mathbb Q$, and no non-isotrivial elliptic curves with good reduction at every place of $\mathbb F_q(t)$, but there are isotrivial examples.

C. Numerical statements tend to be much simpler over function fields.

Szpiro's conjecture over a function field has the form $\Delta= O(N^6)$, not $\Delta= O(N^{6+\epsilon}) as is known to be best possible over number fields. (This was changed from the ABC conjecture to answer Vesselin Dimitrov's objection about the Vojta conjecture being a more natural statement than the ABC conjecture in the function field setting - I think Szpiro's conjecture is also a very natural statement). As Vesselin points out, this might also be related to B, and the fact that the moduli space of elliptic curves is isotrivial.

There exist constructions hitting simple numerical lower bounds, such as extensions of $\mathbb F_q(t)$ of degree $n$ and Galois group $S_n$ whose conductor exactly reaches the lower bound you get by looking at L functions, $q^{2(n-1)}$. One can get similar lower bounds by looking at L functions over number fields, but it is not at all obvious that there exist extensions that reach them.

D. The zeta function has infinitely many poles over function fields, but only one pole over a number field. This makes the ideal-counting and prime-counting functions both logarithmically periodic - i.e. all polynomials, and all prime polynomials, have norm $q^n$ for some $n$, not smoothly distributed like the sizes of numbers and prime numbers are.

One usually deals with this by considering polynomial of fixed degree, and viewing that as the function field analogue of a large interval, but sometimes it recurs in ways that might be surprising. For instance, the error term in the formula for the number of squarefree polynomials of degree $n$ is $2$-periodic in $n$. This sort of makes sense because squaring is $2$-periodic also.

E. The zeta function has finite complexity over function fields, but infinite complexity over number fields. The obvious aspect of this is that the zeros are periodic. Usually one accounts for this by, if considering a problem that involves many zeros, taking the large $g$ limit. However another facet is that each zero is an object with a simple description, being the log of an algebraic number.

This means that phenomena (like two zeros being equal, or a linear dependence among the zeros) that would be infinitely improbable over number fields, and thus we expect that they never happen, unless there is a good reason (like the same L function appearing twice in the product for a Dedekind zeta function, forcing some zeros to occur with multiplicity), are only finitely improbable and thus we expect that to happen occasionally. So the linear independence conjecture, for instance, is known to be false over some function fields, but is known to hold for randomly chosen function fields.

F. There is no infinite place and no $p$-adic Hodge theory over function fields. This causes a number of statements to be simpler - for instance, the analogues of the Fontaine-Mazur conjecture and Langlands conjectures are much simpler.

G. $p$-adic properties, like the Newton polygon of Frobenius, behave much better over function fields, because you don't have to keep changing $p$. For instance, a non-isotrivial elliptic curve over a function field has only finitely many supersingular primes.

H. Over function fields, the Mobius function $\mu(f + g^p)$ is proportional to a quadratic Dirichlet character in $g$ modulo the derivative of $f$. The set of such sums behaves like a short interval / arithmetic progression / Bohr set, in addition to being the set of values of a polynomial, but under none of the special sets is the Mobius function expected to behave like a Dirichlet character over the integers. This underlies the deviation in the Bateman-Horn conjecture mentioned in Lior Bary-Soroker's answer. It also was exploited in recent work of Mark Shusterman (EDIT: and myself).

I. Additive combinatorics seemingly behaved much differently over function fields. Work of Ellenberg and Gijswijt showed that the maximum size of a set of polynomials of degree $<d$ free of three-term arithmetic progressions has size at most $q^d / \left(q^d\right)^\epsilon$ for some $\epsilon>0$ depending on the characteristic. On the other hand, over the integers there are examples due to Behrend of subsets of $\{1,2,\dots,N\}$ free of three-term progressions of size at least $N/ e^ { O(\sqrt \log N)}$. Because $N$ is the analogue of $q^d$, the upper bound in the function field case is much smaller than the lower bound in the number field case, so whatever the true maximum size in each case, the two must be very different.

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  • $\begingroup$ Very thorough, thank you for your insights! $\endgroup$ – Stanley Yao Xiao Jul 18 '15 at 19:34
  • $\begingroup$ Regarding point C on $abc$: The basic case that did not require an $\epsilon$ (i.e., Mason's $abc$ theorem for polynomials, and its extensions to algebraic function elements) is only the split case; it follows immediately from Hurwitz's inequality. The non-split case (e.g., Vojta conjecture for $(\mathbb{P}_{k(t)}^1,[a(t)]+[b(t)]+[c(t)])$ for a non-constant $a,b,c \in k(t)$) is rather more interesting; in char. 0 it was proved by McQuillan and Yamanoi (independently). It does require an $\epsilon$. It is this that should be compared to the arithmetic case, number fields being non-isotrivial. $\endgroup$ – Vesselin Dimitrov Mar 22 '16 at 9:33
  • $\begingroup$ The example I suggested was bad. But if $(X,D)$ is a non-isotrivial pair of an elliptic curve over $k(t)$ and a divisor, then the $abc$ theorem $h_{K_X}(P) + h_D(P) \leq (1+\epsilon) (\mathrm{cond}_D(P)+d(P)) + O_{\epsilon}(1)$ requires an $\epsilon$, and it might be similar to the arithmetic case. Is the error term here (almost) at least as big as the square root from the main term, as is the case for the $abc$ conjecture? $\endgroup$ – Vesselin Dimitrov Mar 24 '16 at 0:11
  • $\begingroup$ @VesselinDimitrov I don't know - this is very far from my area of expertise. In the elliptic case, $K_X(P)$ is bounded, right? And probably you can take $D$ to be a zero section? So $h_D(P)$ is the intersection number of $P$ with the zero section and $\operatorname{cone}_D(P)$ is the number of zeroes. What is $d(P)$ in this context? $\endgroup$ – Will Sawin Mar 24 '16 at 1:08
  • $\begingroup$ $d(P)$ is the logarithmic root discriminant in general. So, in this case, it is the genus of the field generated by $P$, or of the curve corresponding to it (divided by the fibral degree if we are to take the "absolute height"). Sorry, I made a lapsus in writing "elliptic" - I meant the curve to be general. (Otherwise, yes, $K_X$ is irrelevant if the curve is elliptic.) This is the general $abc$ theorem in function fields (when the field has characteristic zero). Before McQuillan and Yamanoi, Vojta could only prove it with the exponent $2+\epsilon$. $\endgroup$ – Vesselin Dimitrov Mar 24 '16 at 1:27
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A Carmichael number is a composite integer $n > 1$ such that $a^{n-1} \equiv 1 \bmod n$ for all integers $a$ relatively prime to $n$. Carmichael numbers can be characterized by Korselt's criterion: a composite integer $n$ is Carmichael iff it is squarefree and for every prime $p$ dividing $n$ we have $(p-1) \mid (n-1)$. Likewise, define a Carmichael polynomial $f(x) \in \mathbf F_q[x]$ to be reducible such that $a^{{\rm N}(f)-1} \equiv 1 \bmod f$ for all $a$ in $\mathbf F_q[x]$ that are relatively prime to $f$, where ${\rm N}(f) = q^{\deg f}$. Korselt's criterion carries over to $\mathbf F_q[x]$: a reducible $f$ in $\mathbf F_q[x]$ is Carmichael if and only if it is squarefree and for every (monic) irreducible $\pi$ dividing $f$ we have $({\rm N}(\pi)-1)\mid ({\rm N}(f)-1)$. This is just reinforcing the analogy between $\mathbf Z$ and $\mathbf F_q[x]$, so what's the point?

A non-analogy is that a Carmichael number can't be a product of two primes, but a Carmichael polynomial can be a product of two irreducibles: the product of two different monic irreducibles of the same degree is a Carmichael polynomial. The reason the analogy is breaking down is that in $\mathbf Z$ two different positive integers don't have the same size, but different monic polynomials in $\mathbf F_q[x]$ can have the same degree. If you were to define Carmichael ideals in number rings, where beyond $\mathbf Z$ there are always different prime ideals of the same norm, a product of two different prime ideals of the same norm is a Carmichael ideal. Thus in a sense the analogy still works as long as you are in a number ring other than $\mathbf Z$, so again what's the point?

The reason I am posting this answer is because of a concrete consequence of Carmichael numbers never being a product of two primes (a fact that by itself looks like an accident) that is an algorithmic non-analogy between $\mathbf Z$ and $\mathbf F_q[x]$. In the Miller-Rabin primality test in $\mathbf Z$ the number of Miller-Rabin witnesses for the compositeness of an odd integer $n$ is always at least 75%, with this lower bound probably being asymptotically sharp (there is a sequence of odd composite $n$ whose proportion of Miller-Rabin witnesses tends to 75% from above if we believe that certain linear expressions can be prime infinitely often together). At one point in the proof of this 75% lower bound it's important to know that a product of two primes is never a Carmichael number. As I said above, the analogue of that fact in $\mathbf F_q[x]$ is false, infinitely often, and as a result in the Miller-Rabin irreducibility test in $\mathbf F_q[x]$ (for odd $q$) the sharpest asymptotic lower bound for the proportion of Miller-Rabin witnesses is less than 75%. This is due entirely to the Carmichael $f$ that are products of two different monic irreducibles of the same degree. For all the other reducible $f$ the proof in $\mathbf Z$ still works and the 75% lower bound goes through. But for a Carmichael polynomial that is a product of two irreducibles of the same degree, the proportion of Miller-Rabin witnesses to reducibility is at most 75% rather than at least 75%, and is in fact strictly less than 75% except a finite number of times. The proportion of Miller--Rabin witnesses for such a polynomial is still always greater than 50%, so it shouldn't take long to find a Miller-Rabin witness for any reducible polynomial, but the usual advantage of the Miller-Rabin test over the Solovay-Strassen test in $\mathbf Z$ based on the higher lower bound on the proportion of witnesses is weakened.

Here is a concrete example. In ${\mathbf F}_7[x]$ let $f(x) = x(x-1)$. The number of Miller-Rabin witnesses is 30 and $30/(7^2-1) = 30/48 = 5/8 = .625 < 3/4$.

For such Carmichael polynomials their proportion of Miller--Rabin witnesses tends to 1/2 or 2/3, rather than 3/4, as the degree of the polynomial tends to infinity. (To have a limit you need to restrict the degree of the irreducible factors in such polynomials to be either even or odd if $q \equiv 3 \bmod 4$.) The asymptotic lower bound on the proportion of witnesses for the Solovay-Strassen test in $\mathbf F_q[x]$ is 50%.

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  • $\begingroup$ This answer is fantastic! I recently taught elementary number theory and included the Miller-Rabin result. This is a marvelously simple demonstration of the difference! $\endgroup$ – Stanley Yao Xiao Jul 18 '15 at 19:33
  • $\begingroup$ @StanleyYaoXiao, if you want to run some numerical tests in $\mathbf F_q[x]$ (say $q = 3$ or $5$) for the Carmichael case of $f = \pi_1\pi_2$ with factors of equal degree, to see how close it comes to 50%, let me know how things turn out. $\endgroup$ – KConrad Jul 20 '15 at 9:15
  • $\begingroup$ @StanleyYaoXiao, I worked out some examples myself now, and included one of them in my edit to the answer. $\endgroup$ – KConrad Jul 28 '15 at 8:37
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Several cases of the Bateman-Horn conjecture on prime values of polynomials are quite different. See the paper by Conrad, Conrad, and Gross in http://math.stanford.edu/~conrad/papers/genuszerofinal.pdf.

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    $\begingroup$ For those who are more inclined toward elliptic curves, it may be of interest to note that the deviation for Bateman-Horn underlies unusual variational behavior of the root number in some non-isotrivial pencils of elliptic curves over global function fields (unlike anything that could occur for non-isotrivial pencils over $\mathbf{Q}$ under standard conjectures); see the paper of Conrad, Conrad, & Helfgott. $\endgroup$ – user27920 Jul 29 '14 at 3:54
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    $\begingroup$ An addendum to user52824's comment: the paper of C, C, and H is on the arXiv at arxiv.org/abs/math/0408153 and a related survey paper, which describes the application to elliptic curves at the end, is math.uconn.edu/~kconrad/articles/texel.pdf. The survey paper may be accessible to a wider audience. $\endgroup$ – KConrad Jul 29 '14 at 9:51

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