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Ordinary Primes for Abelian Surfaces

We compute the density of the set of ordinary primes of an abelian surface over a number field in terms of the l-adic monodromy group. Using the classification of l-adic monodromy groups of abelian surfaces by Fite, Kedlaya, Rotger, and Sutherland, we show the density is 1, 1/2, or 1/4.

The author is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1148900.
1 It does not matter whether we consider connected components of the scheme G or of its geometric form G Q ℓ . Because G is defined to be the Zariski closure of a subset of GSP 4 (Q ℓ ), each connected component of G Q ℓ contains a point of GSP 4 (Q ℓ ) and hence is defined over Q ℓ . n ∈ [−6, 6]. We may compute both a 2 and p in terms of Frob p ∈ GSP 4 (Q ℓ ): the trace of Frob p acting on ∧ 2 V is a 2 , and, because the symplectic form comes from the Weil pairing, the action of Frob p on the similitude character χ is multiplication by p. Hence: Because ∧ 2 V ⊗ χ −1 is an algebraic representation, the set where the trace has a given value is a Zariski closed subset, so is a closed subset in the ℓ-adic topology. Let Z be the finite union over all integers n ∈ [−6, 6] of the closed subset of G where the trace is n. Then Z is a conjugacy-invariant closed subset of G. Let Γ be the image of Gal(Q/K) in GSP 4 (Q ℓ ). Γ is a closed subgroup of GSP 4 (Q ℓ ), hence an ℓ-adic analytic group [Bourbaki, 1975, §8.2 Theorem 2]. Γ is also compact, so it has a Haar measure of total mass one. Z ∩ Γ is an analytic subset of Γ, so its boundary has measure 0 [Serre, 2011, Proposition 5.9]. Hence, by Chebotarev's density theorem, the density of primes lying in Z is equal to the Haar measure of Z ∩ Γ [Serre, 2011, Corollary 6.10]. Because Γ is Zariski dense in G, the Haar measure of Z ∩Γ is equal to the number of connected components of G contained in Z divided by the number of connected components of G [Serre, 2011, Proposition 5.12 and 5.2.1.2].
Thus the density of the set of non-ordinary primes is equal to the number of connected components where the trace is constant and equal to n for some n ∈ [−6, 6] divided by the number of connected components. So it is sufficient to show that on every connected component where the trace of the representation is constant, the trace is equal to one of those 13 values. If there is a connected component where the trace is constant and equal to c, then by Chebotarev again, for infinitely many split primes the trace must equal c. At these primes a 2 is equal to cp. The coefficients of the characteristic polynomial, in particular a 2 , are always integers, so c must be a rational number whose denominator divides p. Because this occurs for infinitely many, hence at least two, different primes p, c is an integer. Then because a 2 ∈ [−6p, 6p], c ∈ [−6, 6]. Therefore, the density of the set of ordinary primes is equal to the proportion of connected components where the trace is nonconstant.
An immediate corollary is: Corollary 2. If G is connected, then the set of ordinary primes has density one.
Proof. By Theorem 1, it is sufficient to show that the trace of ∧ 2 V ⊗ χ −1 is not constant on G. Because the identity matrix is in G, if the trace is constant it is equal to the trace of the identity matrix, namely 6. Hence for every split prime p, we would have a 2 = 6p. But in terms of the four eigenvalues α 1 , α 2 , α 3 , α 4 of Frobenius on H 1 (A, Q ℓ ), a 2 is the sum of the six products α i α j for i < j. As each |α i | = √ p, the only way a 2 can be 6p is if all the eigenvalues are √ p or all are − √ p. But this is impossible, because then a 1 would be 4 √ p or −4 √ p, which is not an integer.
We can make Theorem 1 more explicit using the classification of [Fité, Kedlaya, Rotger, and Sutherland, 2012], which lists all possibilities for the "Sato-Tate group" of an abelian surface. The Sato-Tate group determines the base change of the ℓ-adic monodromy group to Q ℓ . By applying Theorem 1 to each of these monodromy groups, we obtain: Theorem 3. The density of the set of ordinary primes of A is 1 unless A either is a CM abelian surface, or is isogenous to the product of a CM elliptic curve and a non-CM elliptic curve, or is isogenous to the product of CM elliptic curves. In these cases, the density of the set of ordinary primes is: • If A is a CM abelian surface and F is the smallest field that all the endomorphisms of A are defined over, the density is: • If A is isogenous to the product of a CM elliptic curve and a non-CM elliptic curve, the density is 1 if the CM field is contained in K and 1/2 otherwise. • If A is isogenous to the product of two CM elliptic curves with CM fields F 1 and F 2 , the density is: In particular, the density is always 1, 1/2, or 1/4.

Proof.
We can compute the density using Theorem 1 in terms of the action of the ℓ-adic monodromy group on ∧ 2 V ⊗ χ −1 -it is the number of connected components on which the trace is not constant, divided by the number of connected components. This ratio is clearly preserved by extension of scalars from Q ℓ to Q ℓ and from Q ℓ to C, and passage from an algebraic group G/C to the complex Lie group G(C). It is also preserved by passage from a reductive G(C) to a maximal compact subgroup K over C, because a maximal compact subgroup meets each component of a reductive group over C in a Zariski dense subset, the trace on a component of a maximal compact subgroup is constant if and only if the trace on the corresponding component of the complex group is constant.
The group called G 1,Zar ℓ in [Fité et al., 2012, Definition 2.4] is the kernel of the similitude character from the ℓ-adic monodromy group to G m . Because the ℓ-adic monodromy group contains the scalars (by an argument of Deligne, [Serre, 1977, 2.3]), it is equal to G 1,Zar ℓ times the group of scalars. Because the scalars act trivially on ∧ 2 V ⊗ χ −1 , we may as well work with G 1,Zar ℓ . By [Fité et al., 2012, Theorem 2.16], G 1,Zar ℓ is the base change from Q to Q ℓ of a group AST A , which when base changed to C has a maximal compact subgroup ST A . Because the ratio of Theorem 1 is preserved by base change and by passage to maximal compact subgroups (as the ℓ-adic monodromy group is reductive [Faltings, 1986, Theorem 3]), we may as well work with ST A , the "Sato-Tate group".
The group ST A is classified in [Fité et al., 2012, Theorem 4.2] as being one of 52 possible groups. The density of ordinary primes is the fraction of connected components with nonconstant trace on ∧ 2 V , for V the restriction to ST A of the standard representation of USP (4). This makes proving the theorem a process of checking each individual group, which can be split into cases according to the identity component of ST A . The identity component is either USP (4), SU(2) × SU(2), SU(2), U(1) × U(1), SU(2) × U(1), or U(1). A routine computation shows: Case USP (4), SU(2)×SU(2), SU(2): In these three cases, the density is 1. These cases occur when A is a surface with endomorphism group Z, a real multiplication surface, a quaternion multiplication surface, or is isogenous the product of two non-CM elliptic curves.
Case U(1) × U(1): In this case, the density is 1 divided by the number of components. This case occurs when A is isogenous to a simple CM abelian surface or a product of two non-isogenous CM curves. The number of components is equal to the degree of the field extension over which all endomorphisms are defined. In all cases of the [Fité et al., 2012, Theorem 4.2] classification, this degree is either 1, 2, or 4.
Note that there are some subgroups of USP 4 with identity component U(1) × U(1) that are never the monodromy group of an abelian surface, and these ones can have non-identity components with non-constant trace, so their density would not equal 1 divided by the number of components. However, all the groups listed in [Fité et al., 2012, Theorem 4.2] have just one component with non-constant trace.
Case SU (2)×U (1): The density is 1 divided by the number of components. This occurs when A is isogenous to the product of a CM elliptic curve with a non-CM elliptic curve. The number of components is 1 if the CM field is contained in K and 2 otherwise.
Case U(1): This occurs when A is geometrically isogenous to a product of two copies of the same CM elliptic curve. The surface is ordinary at a prime if and only if that curve is ordinary, so the density of ordinary primes is 1 if the CM field is contained in K and 1/2 otherwise. These facts are summarized by the statement of this theorem.