# Frobenius Tori

I've been trying to understand a fragment of a letter of Serre to Ken Ribet. In this letter, Serre describes the construction of the Frobenius Torus associated to an abelian variety defined over a finite field.

I decided to translate it here for future reference; shoutout to Google translate, and apologies to professor Serre for butchering his words. The letter in question is article 133 from Serre's collected papers, volume IV: Lettres à Ken Ribet du 1/1/1981 et du 29/1/1981. More specifically, I am interested in pages 6 to 11.

Of course, all errors (mathematical and literary) are my fault.


Here it goes:

 4. Frobenius Tori (finite fields)

First, let $$A$$ be an abelian variety over a finite field $$k$$ with $$q$$ elements, and let $$\pi$$ be its Frobenius endomorphism. We can attach to $$\pi$$ a certain group of multiplicative type $$\Theta_\pi$$, where the neutral component $$T_\pi$$ is a torus - all these groups are defined over $$\mathbf{Q}$$. This is how it goes: we first consider the commutative semisimple algebra $$\mathbf{Q}(\pi)$$ generated by $$\pi$$; we can, if we wish, decompose it as a product of fields $$K_i$$; the "multiplicative group of $$\mathbf{Q(\pi)}$$" is a $$\mathbf{Q}$$-torus $$T_{\mathbf{Q}(\pi)} = \prod T_{K_i}$$, where the group of $$\mathbf{Q}$$-points is $$\mathbf{Q}(\pi)^\times$$. We define $$\Theta_\pi$$ as the smallest algebraic subgroup of $$T_{\mathbf{Q}(\pi)}$$ containing $$\pi$$. This group is not necessarily connected. Its neutral component is $$T_\pi$$.

If $$n = \dim A$$, the algebra $$\mathbf{Q}(\pi)$$ possesses a $$\mathbf{Q}$$-module $$V_\mathbf{Q}$$ of rank $$2n$$ who by extension of scalars to $$\mathbf{Q}_\ell$$, gives the Tate modules $$V_\ell$$, seen as modules over $$\mathbf{Q}_\ell(\pi)$$. (I don't believe there is a "natural" construction of this module $$V_\mathbf{Q}$$. Its existence comes, if you will, from the fact that $$\mathbf{Q}(\pi)$$ is commutative, and that the trace of its representation in $$V_\ell$$ is defined over $$\mathbf{Q}$$, and independent of $$\ell$$.) If $$A = \prod A_i$$ is the decomposition of $$A$$ corresponding to that of $$\mathbf{Q}(\pi) = \prod K_i$$, we have $$V_\mathbf{Q} = \oplus V_{\mathbf{Q},i}$$ where $$V_{\mathbf{Q},i}$$ is a $$K_i$$-vector space of dimension $$2\dim A_i \, / \, [K_i:\mathbf{Q}]$$.

This gives a representation (over $$\mathbf{Q}$$) of the groups $$\Theta_i$$ and $$T_\pi$$, of dimension $$2n$$. This representation identifies these groups with subgroups of $$\mathsf{GL}_{2n}$$ (and even the group $$\mathsf{GSp}_{2n}$$ of symplectic similitudes, once a polarization of $$A$$ is chosen.) The eigenvalues $$\lambda_1, \dots, \lambda_{2n}$$ of $$\pi$$ are the usual "Frobenius eigenvalues". If we write $$\pi$$ in diagonal form (after a convenient extension of scalars), we find that $$\Theta_\pi$$ is the group of diagonal matrices $$(t_1, \dots, t_{2n})$$ such that $$\prod t_i^{m_i} = 1$$ for any tuple of integers $$(m_i)$$ such that $$\prod \lambda_i^{m_i}=1$$ (cf. Chevalley, Theorie des Groupes de Lie, tome II, Hermann 1982, chap. II, fin du § 13); an analogous definition for $$T_\pi$$ is obtained by replacing the condition "$$\prod \lambda_i^{m_i} = 1$$" by "$$\prod \lambda_i^{m_i}$$ is a root of unity''. The character group of $$\Theta_\pi$$ is identified (action of $$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$$ included), with the subgroup of $$\overline{\mathbf{Q}}^\times$$ generated by the $$\lambda_i$$; dividing this group by its torsion subgroup, we find the character group of the torus.

Thanks to Tate, we know what the possible $$\pi$$'s are, and therefore we know in principle what the tori $$T_\pi$$ are . I say "in principle" because, in practice, it is not as trivial as that. Here are some examples:

• ($$\dim A = 1$$) If the elliptic curve is supersingular, we have $$T_\pi = \mathbf{G}_m$$, which identifies with the group of homotheties $$\begin{bmatrix} t & 0 \\ 0 & t \end{bmatrix}$$. If the curve is ordinary, $$T_\pi$$ is the torus $$T_K$$ attached to an imaginary quadratic field $$K$$, and over $$\overline{\mathbf{Q}}$$ this torus becomes the diagonal torus $$\begin{bmatrix} t_1 & 0 \\ 0 & t_2 \end{bmatrix}$$ of $$\mathsf{GL}_2$$.
• ($$\dim A = 2$$) Unless I am mistaken, there are four possibilities:
• Case 2.1: ($$\dim T_\pi = 1$$) The torus $$T_\pi$$ is the torus $$\mathbf{G}_m$$ of homotheties; it consists of diagonal matrices $$(t_1, \dots, t_4)$$ such that $$t_1 = t_2 = t_3 = t_4$$. (Example: a product of two supersingular elliptic curves).
• Case 2.2: ($$\dim T_\pi = 2$$) After an extension of scalars, the equations of $$T_\pi$$ are $$t_1 = t_2$$ and $$t_3 = t_4$$. (Example: the product of two isogenous ordinary elliptic curves.)
• Case 2.3: ($$\dim T_\pi = 2$$) After an extension of scalars, the equations of $$T_\pi$$ are $$t_2 = t_3$$ and $$t_2^2 = t_1t_4$$. (Example: The product of a supersingular with an ordinary curve.)
• Case 2.4: ($$\dim T_\pi = 3$$) After an extension of scalars, the equation of $$T_\pi$$ is $$t_1t_4 = t_2t_3$$, the maximal torus of the group $$\mathsf{GSp}_4$$ (I've chosen the notation in so that the previous ones are contained in this one). (Example: the product of two non-isogenous ordinary elliptic curves).

I have given these examples to make the following result plausible to you; it will play a major role on the future.

For a fixed $$n$$, the tori $$T_\pi$$ belong to a finite number of conjugacy classes in $$\mathsf{GL}_{2n}$$ (geometric conjugation, i.e., over $$\overline{\mathbf{Q}}$$).

For $$n=1$$ (respectively $$n=2$$), this number is equal to $$2$$ (respectively $$4$$).

Before proving the finiteness theorem above, I need properties of the torus $$T_\pi$$. First of all, as Deligne remarked, $$T_\pi$$ contains the group $$\mathbf{G}_m$$ of homotheties (corresponding to $$\mathbf{Q}^\times$$ embedding into $$\mathbf{Q}(\pi)^\times$$ – and also corresponding to the homotheties of $$\mathsf{GL}_{2n}$$). Indeed, since all the $$\lambda_i$$ have the same archimedian absolute value, every relation $$\prod_i \lambda_i^{m_i} =$$ "root of unity" leads to $$\sum m_i =0$$, and the diagonal matrix $$(t,\dots,t)$$ satisfies the relation $$t^{\sum m_i} = 1$$.

(Also note that there is a canonical homomorphism $$N: T_\pi \to \mathbf{G}_m$$ (and even $$\Theta_\pi \to \mathbf{G}_m$$) coming from $$\Theta_\pi \to \mathsf{GSp}_{2n} \to \mathbf{G}_m$$, and characterized by $$N(\pi) = q$$. The composition $$\mathbf{G}_m \to T \to \mathbf{G}_m$$ is simply $$x \mapsto x^2$$; the situation is analogous to that of McGill, II-32, excerc. 1.)

After the archimedian absolute values, you have to look at the absolute values (or rather the valuations) $$v$$ above $$p$$ (the other ones evidently give you nothing, since all the $$\lambda_i$$ are units). Let $$v$$ be a valuation of $$\overline{\mathbf{Q}}$$ above $$p$$. Let $e_i = v(\lambda_i)/v(q) \quad (i = 1, \dots, 2n),$ the $$\lambda_i$$ being as above the eigenvalues of $$\pi$$ in its natural representation. If $$\prod \lambda_i^{m_i}$$ is a root of unity, its $$v$$ valuation is 0, and therefore we have $$\sum e_i m_i = 0$$. If $$d$$ is a common denominator of the $$e_i$$, the 1-parameter group $t \mapsto (t^{de_1}, \dots, t^{de_{2n}})$ is contained in the torus $$T_\pi$$. (Off course, this group is only defined over $$\overline{\mathbf{Q}}$$; more precisely the natural action of $$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$$ on one parameter subgroups of $$T_\pi$$ transitively permutes these subgroups - since this action permutes the $$v$$ amongst them).

To exploit this, it is convenient to introduce the group $$Y_\pi$$ of "1-parameter subgroups" or "cocharacters" of $$T_\pi$$ (the dual of the character group), as well as $$Y_{\pi, \mathbf{Q}} = \mathbf{Q}\otimes Y_\pi$$. We can reformulate the previous construction by saying that the $$(e_1, \dots, e_{2n})$$ define an element $$y_{v,\pi}$$ of $$Y_{\pi,\mathbf{Q}}$$.

The $$\mathbf{Q}$$-vector space $$Y_{\pi,\mathbf{Q}}$$ is generated by the conjugates of $$y_{v,\pi}$$.

(These are the conjugates of $$y_{v,\pi}$$ by the action of $$\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$$, of course.)

Denote by $$y_{\infty,\pi}$$ the element $$(1,\dots,1)$$ of $$Y_{\pi}$$ (the one that corresponds to the subgroup $$\mathbf{G}_m$$ of homotheties). If $$c \in \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$$ is a complex conjugation, we know that $$\lambda_i\lambda_i^c = q$$. We conclude that

\begin{equation*} v(\lambda_i) + (cv)(\lambda_i) = v(q), \end{equation*}

from where conclude that

\begin{equation*} y_{v,\pi} + y_{cv,\pi} = y_{\infty,\pi}, \end{equation*}

which shows that $$y_\infty$$ belongs to the subspace $$Y'$$ of $$Y_{\pi, \mathbf{Q}}$$ generated by the conjugates of $$y_{v, \pi}$$. To show that that the space $$Y'$$ is in fact equal to $$Y_{\pi, \mathbf{Q}}$$, it suffices, by duality, to prove this: If $$(m_i)$$ is a family of integers orthogonal to all $$y_{v,\pi}$$, then $$\prod_i \lambda_i^{m_i}$$ is a root of unity. Now, let $$(m_i)$$ be such a family, and let $$\lambda = \prod_i \lambda_i^{m_i}$$. From the fact that $$(1,\dots,1)$$ belongs to $$Y'$$, all the archimedian absolute values of $$\lambda$$ are equal to $$1$$; because the $$y_{v,\pi}$$ belong to $$Y'$$, the same is true of the ultrametric absolute values above $$p$$; finally, the same is trivially true for the ultrametric absolute values elsewhere than $$p$$. By a well-known lemma, this implies that $$\lambda$$ is a root of unity.

On the other hand, we have:

The families $$(e_1,\dots,e_{2n})$$, for a given $$n$$, are finite.

In fact, we know that the $$e_i$$ are rational, between $$0$$ and $$1$$, and the denominator of each of them is $$\leq 2n$$. (This can be seen, either by the determination done by Tate of the possible Frobenii of a given abelian variety, or as Fontaine pointed out to me, by the integrality properties of the Dieudonné module.)

For example, if $$n=2$$, the Newton polygon giving the $$e_i$$ is necessarily one of the following three types:

Case (i) corresponds to case 2.1. Case (ii) corresponds to case 2.3 if the "only" conjugate of $$y_v$$ is $$(1, \tfrac 12, \tfrac 12, 0)$$ and to case 2.4 if $$y_v$$ has other conjugates. Case (iii) corresponds to 2.2 if the only conjugate of $$y_v$$ is $$(1,1,0,0)$$, and to 2.4 if there are any others.

I hope that, now, the finiteness theorem is evident; it follows from the two theorems above, since they show that the subspace $$Y_{\pi, \mathbf{Q}}$$ of $$\mathbf{Q}^{2n}$$ has only a finite number of possible positions.

Remark – One could do analogous things with the étale cohomology of a smooth projective variety, in a given dimension $$r$$. In fact, we only really used the following results:

a) The characteristic polynomial of Frobenius $$\pi$$ has integer coefficients (and it is independent of $$\ell$$);

b) The eigenvalues $$\lambda_i$$ of $$\pi$$ have absolute value $$q^{r/2}$$ at the archimedian places, and $$1$$ at the non-archimedian places not divisible by $$p$$;

c) if $$v$$ is a place above $$p$$, the $$e_i = v(\lambda_i)/v(q)$$ take only a finite number of possible values: they are rational numbers between $$0$$ and $$r$$, and the denominators are $$\leq B$$, where $$B$$ is the Betti number of the variety in dimension $$r$$.

(This last property follows from the fact that $$\pi$$ can also be seen as an endomorphism of the crystalline cohomology of the variety. See for example Mazur's exposition at Arcata, p.250, where unfortunately Mazur assumes the variety can be lifted to characteristic 0 over the Witt vectors. This is also between the lines of Berthelot's exposition at Arcata – and this was guaranteed to me by Fontaine!)

Emacs 27.2 (Org mode 9.4.4)