# $\mathbb{F}_1$ and the ABC-conjecture

Finally, we’re closing in on Smirnov’s approach to the ABC-conjecture via geometry over the field with one element.

The geometric defect

Let $\phi : C_1 \mapsto C_2$ be a cover of curves defined over $k$, then the scheme-version of the Riemann-Hurwitz inequality is

$$2 g_{C_1} – 2 \geq deg(\phi) (2 g_{C_1} -2) + \sum^{scheme}_{P \in C_1} (e_{\phi}(P)-1) deg(P)$$

In the special case, when $f$ is a non-constant rational function in $k(C)$ and $f~:~C \mapsto \mathbb{P}^1_k$ is the corresponding cover, this reads

$$2g_C-2 \geq -2 deg(f) + \sum^{scheme}_{P \in C} (e_f(P)-1) deg(P)$$

which can be turned into the inequality

$$\sum^{scheme}_{P \in C} \frac{(e_f(P)-1) deg(P)}{deg(f)} \leq 2 – \frac{2-2g_C}{deg(f)}$$

We call the expression $\delta(P) = \tfrac{(e_f(P)-1) deg(P)}{deg(f)}$ the defect of $P$. Observe that $\delta(P) \geq 0$ and so this inequality only improves it we restrict the summation to some subset of schematic $C$-points.

The arithmetic defect

Take a positive rational number $q = \frac{m}{n}$ with $1 \leq n < m$ and $(m,n)=1$ and consider the cover

$$q~:~\mathsf{Spec}(\mathbb{Z}) \mapsto \mathbb{P}^1 / \mathbb{F}_1$$

Recall that the fiber over the point $[d] \in \mathbb{P}^1 / \mathbb{F}_1$ consists of all prime divisors of $m^d-n^d$ not dividing any $m^e-n^e$ for $e < d$. The fiber of $[0]$ (resp. of $[\infty]$) consists of all prime divisors of $m$ (resp. of $n$ together with $\infty$). Here's part of the cover for $q=\frac{104348}{33215}$ (a good rational approximation for $\pi$).

It is tempting to define the ramification index $e_q(p)$ for the map $q$ in the prime $p$ lying in the fiber $q^{-1}([d])$ to be the largest power of $p$ dividing $m^d-n^d$. Likewise, for $p \in q^{-1}([0])$ (resp. in $q^{-1}([\infty])$) take for $e_q(p)$ the largest power of $p$ dividing $m$ (resp. dividing $n$). Finally, take $e_q(\infty) = log(q)$.

Combine this with our previous definitions for the degree of $p$ to be $log(p)$ and of the degree of the map $q$ to be $log(m)$, to define the arithmetic defect of $q$ in the prime $p$ to be

$$\delta(p) = \frac{(e_q(p)-1) log(p)}{log(m)}$$

We can now define the total defect of the cover $q$ over the point $[d] \in \mathbb{P}^1 / \mathbb{F}_1$ to be

$$\delta_{[d]} = \sum_{p \in q^{-1}([d])} \delta(p)$$

It is easy to work out these total defects for the four $\mathbb{F}_1$-rational points of $\mathbb{P}^1 / \mathbb{F}_1$ : $\{ [0],[1],[2],[\infty] \}$ (the primes lying on the blue lines in the graph).

For a natural number $a$ let $a_0$ be its square-free part and $a_1 = \tfrac{a}{a_0}$ the remaining part. Then

• $\delta_{[0]} = \frac{log(m_1)}{log(m)}$
• $\delta_{[\infty]} = \frac{log(n_1)+log(q)-1}{log(m)}$
• $\delta_{[1]} = \frac{log((m-n)_1)}{log(m)}$
• $\delta_{[2]}= \frac{log(k_1)}{log(m)}$

where $k$ is $m+n$ divided by the largest $2$-power it may contain.

Hurwitz-conjecture for $\mathbb{Q}$

If we sum the defects of $q$ in all primes over the points $\{ [0],[1],[\infty] \}$ we would get, in analogy with the Hurwitz-inequality in the function field case

$$\delta_{[0]}+\delta_{[1]}+\delta_{[\infty]} \leq 2 – \frac{2 – 2g_{\mathsf{Spec}(\mathbb{Z})}}{log(m)}$$

We do not know what the genus of the arithmetic curve $\mathsf{Spec}(\mathbb{Z})$ might be, but is sure is a constant not depending on the map $q$. If we could develop a geometry over $\mathbb{F}_1$ such that all wild guesses we made before would turn out to be the correct ones for an $\mathbb{F}_1$-version of the Hurwitz inequality, we would have the statement below :

For every $\epsilon > 0$ there exists a constant $C(\epsilon)$ such that the following inequality holds for every pair $1 \leq m < n$ with $(m,n)=1$

$$\frac{log(m_1) + log((m-n)_1) + log(n_1) + log(m)-log(n)-1}{log(m)} \leq 2 + \epsilon + \frac{C(\epsilon)}{log(m)}$$

‘Proof’ of the ABC-conjecture

The ABC-conjecture requires for every $\epsilon > 0$ a constant $D(\epsilon)$ such that for all coprime natural numbers $A$ and $B$ we have with $A+B=C$

$$C \leq D(\epsilon) (A_0B_0C_0)^{1+\epsilon}$$

Well, take $m=C$ and $n=min(A,B)$ then in the conjectural Hurwitz inequality for the cover corresponding to $q=\frac{m}{n}$ above we have that

• $\frac{log(m_1)}{log(m)} = 1 – \frac{log(m_0)}{log(m)}$
• $\frac{log(n_1)+log(m)-log(n)-1}{log(m)} = 1 – \frac{log(n_0)}{log(m)} – \frac{1}{log(m)}$
• $\frac{log((m-n)_1)}{log(m)}=\frac{log(m-n)}{log(m)}-\frac{log((m-n)_0)}{log(m)} \geq 1 – \frac{log((m-n)_0)}{log(m)} – \frac{log(2)}{log(m)}$

(the latter inequality because $m-n \geq \frac{m}{2}$ and so $log(m-n) \geq log(m)-log(2)$). Plug this into the inequality above and get

$$3-\frac{log(n_0m_0(m-n)_0)}{log(m)} \leq 2 + \epsilon + \frac{C(\epsilon) + 1 + log(2)}{log(m)}$$

Take $log(C'(\epsilon))=C(\epsilon)+1+log(2)$ and reshuffle in order to get the inequality $m^{1-\epsilon} \leq C'(\epsilon)(n_0m_0(m-n)_0)$. But then, finally (finally!) with $D(\epsilon)=C'(\epsilon)^{1+\epsilon}$

$$C=m \leq D(\epsilon)(n_0m_0(m-n)_0)^{1+\epsilon} = D(\epsilon)(A_oB_0C_0)^{1+\epsilon}$$

# ABC-theorem for Curves

Here we give the promised proof of the ABC-conjecture for function fields.

As always, $k$ is a perfect (e.g. finite) field and $K=k(X)$ is the function field of a smooth projective curve $X$ defined over $k$. We take elements $u,v \in K^*$ satisfying $u+v=1$ and consider the cover $u : X \mapsto \mathbb{P}^1_k$ corresponding to the embedding $k(u) \hookrightarrow K$. We want to determine the (schematic) zero- and pole-divisors of $u$ and $v$ and call them $A=div_0(u), B=div_0(v)$ and $C=div_{\infty}(u)=div_{\infty}(v)$.

Let $R$ be the integral closure of $k[u]$ in $K$, then in $R$ we can write the ideals $u$ and $(v)=(1-u)$ as products of prime-ideals (which correspond to schematic points of $X$)

$(u) = P_1^{e_u(P_1)} \cdots P_r^{e_u(P_r)}$
$(v) = Q_1^{e_u(Q_1)} \cdots Q_s^{e_u(Q_s)}$

and so $A = \sum_i e_u(P_i) [P_i]$ and $B = \sum_j e_u(Q_j) [Q_j]$. If $S$ is the integral closure of $k[\frac{1}{u}]$ in $K$, then we have in $S$ a decomposition

$(\frac{1}{u}) = R_1^{e_u(R_1)} \cdots R_t^{e_u(R_t)}$

and therefore $C = \sum_l e_u(R_l)[R_l]$. We already know that $deg(A)=deg(B)=deg(C)=n=[K : k(u)]$.

Case 1 : Let us assume that the field extension $K/k(u)$ is separable. Then, by the Riemann-Hurwitz formula (or rather, the scheme-version of it) we get the inequality (use that the genus of $\mathbb{P}^1_k$ is zero) :

$2 g_K – 2 \geq -2n + \sum^{scheme}_{P \in C} (e_u(P)-1) deg(P)$

Because for all points $e_u(P)-1 \geq 0$, the inequality only becomes better if we restrict the sum to a subset of points, say to the support of $A+B+C$ (that is to ${ P_1,\cdots,P_r,Q_1,\cdots,Q_s,R_1,\cdots,R_t }$). Then we get

$2 g_K -2 \geq -2n + \sum_{P \in Supp(A+B+C)} e_u(P)deg(P) – \sum_{P \in Supp(A+B+C)}deg(P)$
$~\qquad = -2n+3n-\sum_{P \in Supp(A+B+C)} deg(P)$

which gives us the required form of the ABC-conjecture for curves

$n=deg(u)=deg_s(u) \leq 2g_K – 2 + \sum_{P \in Supp(A+B+C)} deg(P)$

Case 2 : If $K/k(u)$ is not separable, take a maximal separable subfield $k(u) \subset M \subset K$, then by definition of $deg_s(u)$ and case 1 we have

$deg_s(u) \leq 2g_M – 2 + \sum_{P’ \in Supp(A’+B’+C’)} deg(P’)$

where $A’$ (resp. $B’$,$C’$) are the schematic fibers of the cover $Y \mapsto \mathbb{P}^1_k$ over the $k$-rational points $0$ (resp. $1$, $\infty$) and where $Y$ is the curve with function field $M$. We are done if we can show that $g_K=g_M$ and that in the cover $X \mapsto Y$ there is a unique point $P$ lying over each point $P’$ with $deg(P)=deg(P’)$.

As $K/M$ is purely inseparable, we have a tower of subfields

$M=M_0 \subset M_1 \subset \cdots \subset M_z=K$

such that $M_i / M_{i-1}$ is purely inseparable of degree $p$ for all $i$. That is, raising to the $p$-th power gives a field-isomorphism $M_i \simeq M_{i-1}$. The genus is a field-invariant, so $g_{M_i}=g_{M_{i-1}}$ and there is a bijection between the dvr’s in $M_i$ and $M_{i-1}$. That is, a bijection between points $P_i \leftrightarrow P_{i-1}$ of the corresponding curves $Y_i \mapsto Y_{i-1}$. Finally, because $t_{P_i}^p = t_{P_{i-1}}$ it follows that $deg(P_i)=deg(P_{i-1})$, and we are done by induction on $i$.

# The ABC-conjecture

In 1985 Joseph Oesterle (left) and David Masser (right) formulated the conjecture that for three relative prime integers satisfying $A+B=C$, the product of the prime divisors of $ABC$ is rarely much smaller than $C$.

More precisely, if $A,B,C \in \mathbb{Z}$ are such that $A+B=C$ and $\gcd(A,B,C)=1$, then their conjecture states that for each $\epsilon > 0$ there is a constant $M_{\epsilon}$ such that for all triples $(A,B,C)$ satisfying the conditions we have

$$\max( |A|, |B|, |C|) \leq M_{\epsilon}\left(\underset{p | ABC}{\prod} p\right)^{1+\epsilon}$$

The ABC-conjecture has several consequences, some obvious ones such as proving Fermat’s last theorem for large exponents, some less obvious such as Falting’s theorem. However, many people consider a proof the ABC-conjecture to be beyond the range of the available methods.

Since 2006 the ABC@Home project tries to find triples $(A,B,C)$ of large ‘quality’ meaning that the ratio

$$\operatorname{q}(A,B,C) = \frac{\log(C)}{\log(\operatorname{rad}(ABC))}$$

is as large as possible. To date, the champion-triple is $2+3^{10}109 = 23^5$ (discovered by Eric Reyssat) with a quality of $1.6299$.

If we write $u = \tfrac{A}{C}$ and $v=\tfrac{B}{C}$ then the ABC-conjecture can be recast as the statement that there is a constant $M_{\epsilon}$ such that when $u,v \in \mathbb{Q}^*$ satisfy $u+v=1$ we have

$$\max\left(\operatorname{ht}(u),\operatorname{ht}(v)\right) \leq M_{\epsilon} + (1+\epsilon)\left(\sum_{p | ABC} \log(p)\right)$$

where $A$ and $B$ are the numerators of $u$ and $v$ and $C$ is their common denominator, and where the ‘height’ $\operatorname{ht}(u)$ of a rational number $u=\tfrac{A}{C}$ with $(A,B)=1$ is $\max\left(\log|A|, \log|C|\right)$.

The latter formulation can be extended to the case of function fields of curves. So, let $K \in \mathsf{1Fields}$ with a perfect field of constants $k$ and suppose $u,v \in K^*$ are non-constants satisfying $u+v=1$. We need a substitute for the notion of height.

If $L$ is the maximal separable extension of $k(u)$ in $K$, then we call the dimension $[L : k(u)]$ the separability degree of $u$ and denote it with $\deg_s(u)$. Clearly, $\deg_s(u) \leq \deg(u) = [K : k(u)]$.

If $R$ is the integral closure of $k[u]$ in $K$, then there are maximal ideals $P_i$ in the Dedekind domain $R$ such that

$$(u) = P_1^{e_1} \cdots P_r^{e_r}$$

Because the local ring in $P_i$ is a discrete valuation ring in $K$ it determines a point in the curve $C$ with $K=k(C)$ (see here) also denoted $P_i$. But then, the zero-divisor of $u$ is $\operatorname{div}_0(u) = A = \sum_i e_i [P_i]$ with degree $\deg(A) = \sum_i e_i \deg(P_i)$.

Similarly, in the integral closure $S$ of $k[\tfrac{1}{u}]$ we have a decomposition

$$(\tfrac{1}{u}) = Q_1^{f_1} \cdots Q_s^{f_s}$$

and the pole-divisor of $u$ is $\operatorname{div}_{\infty}(u) = C = \sum_j f_j [Q_j]$ with degree $\deg(C) = \sum_j f_j \deg(Q_j)$. With these conventions, the ABC-conjecture for function fields can now be formulated as the following claim:

Let $K \in \mathsf{1Fields}/k$ and $u,v \in K^*$ with $u+v=1$, then

$$\deg_s(u) = \deg_s(v) \leq 2 g_K – 2 + \sum_{P \in \operatorname{Supp}(A+B+C)} \deg(P)$$

where $A=\operatorname{div}_0(u)$, $B=\operatorname{div}_0(v)$, $C=\operatorname{div}_{\infty}(u)=\operatorname{div}_{\infty}(v)$ and $g_K$ is the genus of $C$. Observe that there is no $\epsilon$ in this function field ABC-conjecture.

Perhaps surprisingly, the function-field ABC-conjecture can be proved fairly easily from the Riemann-Hurwitz genus formula. Details are in the book Number Theory in Function Fields by Michael Rosen (theorem 7.17) or in an upcoming prep-notes post.

# The Smirnov letters

In the paper The notion of dimension in geometry and algebra, Yuri I. Manin writes :

“This guess involves the conjectural existence of a geometrical world defined over “an absolute point” $\mathsf{Spec}~\mathbb{F}_1$ where $\mathbb{F}_1$ is a mythical field with one element. For some insights about this world, see [J. Tits, Sur les analogues algebriques des groupes semi–simples complexes], [A. Smirnov, Hurwitz inequalities for number fields], [A. Smirnov, Letters to Yu. Manin of Sept. 29 and Nov. 29, 2003], [M. Kapranov, A. Smirnov, Cohomology determinants and reciprocity laws: number field case], [Yu. Manin, Three–dimensional hyperbolic geometry as ∞–adic Arakelov geometry], [C. Soule, Les varietes sur le corps a un element].”

The first of two letters from Alexandr Smirnov to Manin also appears in the paper Lambda-rings and the field with one element by James Borger :

“The second purpose is to prove the Riemann hypothesis. With the analogy between integers and polynomials in mind, we might hope that $\mathsf{Spec}~\mathbb{Z}$ would be a kind of curve over $\mathsf{Spec}~\mathbb{F}_1$ , that $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$ would not only make sense but be a surface bearing some kind of intersection theory, and that we could then mimic over $\mathbb{Z}$ Weil’s proof of the Riemann hypothesis over function fields. The origins of this idea are unknown to me. Manin [Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa)] mentions it explicitly. According to Smirnov [Alexandr L. Smirnov, Letter to Y. Manin. September 29, 2003], the idea occurred to him in 1985 and he mentioned it explicitly in a talk in Shafarevich’s seminar in 1990. It may well be that a number of people have had the idea independently since the appearance of Weil’s proof.”

I thank Yuri I. Manin and James Borger for providing me with copies of these letters. Some of their content is crucial to understand the genesis of Smirnov’s paper Hurwitz inequalities for number fields, which is the first paper we will study in the seminar.

From Smirnov’s letter to Yu. I. Manin, dated september 29th 2003 (links added):

“As to my investigations, I work on the topic since 1985. Till then the subject has been paid next to no attention, with the exception of a paper by J. Tits (1957) and the idea (due to D. Quillen?) to interpret the Barrat-Priddy-Quillen Theorem as the equality $K(\mathbb{F}_1) = \pi^{st}(S^0)$.

My initial goal was (and still is) to construct a “world” which contains algebraic geometry as well as arithmetic, and where all constructions from algebraic geometry (including $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$) would be available. From the beginning I believed that this could give an approach to the Riemann hypothesis (similar to Weil’s approach).

I started with the idea, known for me from a seminar (early 80-s), that sets can be considered as vectorspaces over $\mathbb{F}_1$. I believed that this idea was promising in view of the mentioned interpretation of the Barrat-Priddy-Quillen Theorem.

Since I couldn’t invent $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$, I worked out a strategy which I have adhered:

• “If we can’t develop the whole desired theory, we should invent as many objects over $\mathbb{F}_1$ as possible and establish connections between them.”
• “Since the situation is extremely rigid, any flexibility of constructions would lead to essential progress.”

The idea to construct finite extensions of $\mathbb{F}_1$ (thus getting the missing flexibility) and the suggestion to consider the monoids $0 \cap \pmb{\mu}$ as a naive technical approximation to these extensions arose precisely from this strategy.

Having on hand the extensions, I discovered I could effectively work with a number of new objects over $\mathbb{F}_1$, for instance with $\mathbb{P}^n$. Thus I decided to handle intersection theory (which is part of Weil’s approach to the Riemann hypothesis) on the surface

$$\mathbb{P}^1/\mathbb{F}_1 \times \mathsf{Spec}~\mathbb{Z}$$

instead of the more complicated $\mathsf{Spec}~\mathbb{Z} \times \mathsf{Spec}~\mathbb{Z}$. The Hurwitz genus formula for a map of curves $f~:~X \rightarrow Y$ can be viewed as an example of using intersection theory on the surface $X \times Y$, and I started with it. I succeeded in stating a certain approximation to the Hurwitz formula for the “map”

$$f~:~\mathsf{Spec}~\mathbb{Z} \rightarrow \mathbb{P}^1/\mathbb{F}_1 \qquad \text{where f \in \mathbb{Q}}$$

It was somewhat surprising (and confirming the importance of the approach) that this approximation gave very profound assertions like the ABC-conjecture and others. An impulse to publish these results was given to me by M. Kapranov after he and V. Voevodsky learned about them (1988 or 1989).”

# noncommutative geometry seminar 2011

## what is #angs@t?

Welcome to #angs@t (pronounced ‘angst’), the group-blog accompanying the master-course ‘seminar noncommutative geometry’ given at Antwerp University (Belgium).

Clearly, ‘angs’ is short for Antwerp Noncommutative Geometry Seminar, and the addendum @t indicates that all tweets about the seminar should include the hashtag #angs. Such tweets will appear in the sidebar on the main page.

## practical info

The IRL-part of the seminar begins every friday around 13h in room G 0.16 (campus Middelheim) and ends sometime after 16h when exhaustion strikes lecturer and/or public. Normally lectures are given in Dutch unless the assembled public demands otherwise, in which case we effortless switch to pidgin English.

The virtual-part of the seminar happens here and on twitter, and will be entirely in English. Here, we will collaboratively try to write course-notes, using the EditFlow plugin. Anyone interested to contribute can send an email to lieven.lebruyn at ua.ac.be. If you have a twitter-account, please tweet about the seminar using the hashtag #angs.

# the plan

We will try to sketch an approach to the next biggest conjecture in number theory (after the Riemann hypothesis), the ABC-conjecture, using geometry over the field with one element $\mathbb{F}_1$ and noncommutative geometry.

Here’s a crude outline of the topics/papers we will cover (anticipate major changes):