We also prove that the set of abelian varieties satisfying these conditions is dense in a corresponding modular space. In particular, the set of simple abelian varieties with bad reductions with fixed dimension of the toric parts is dense. The focus of my work is on the Mumford-Tate conjecture for special classes of abelian varieties.
MT for abelian varieties of CM type is a consequence of the results of Shimura and Taniyama [ShT], [Po]. Specifically, let M be the module space of abelian varieties with fixed polarization, level structure and an action k (cf. 11.2). In particular, the set of simple abelian varieties having fixed dimension bad reductions of toric parts is dense.
Although non-CM abelian varieties with (potentially) good reduction exist everywhere (e.g. [So]), this is a very rare occasion. For example, if A is a ~belian variety with End0(A) = Q and the dimension of the toric part of its reduction, which is not necessarily equal in this case, is prime to dim(A), then MT holds. 2) MT holds for abelian varieties of the Ribet type (Theorem 1.2).
Basics, generalities, first applications
Proposition (Mumford): The Hodge group Hg(A) is the largest (reductive) subgroup of GL(V) that fixes all Hodge classes of k', s ~ 1. Since every abelian variety over C has a pattern over a finitely generated field, the theorem implies that Hodge's conjecture is a consequence of Tate's conjecture (conveniently stated, cit. loc.). If the abelian variety is of type I (resp. II, resp. III), then the irreducible components are symplectic (resp. symplectic, resp. orthogonal) (loc.cit). 0.3.2 Assertion: the adaptation and i)t are semisimple if and only if the signature of the action k is ( m, m.
If the signature of the k-action is an Abelian variety of type IV ( m, m ), such an Abelian variety is called an Abelian variety of Weil type (cf. [W]. Sen ([Se), Section 4, Theorem 1 ), the Hodge-Tate decomposition on V l can connect cocharacter. Note: We can see that this cocharacter is exactly analogous to the J.L cocharacter associated with the Hodge decomposition on Vc, Gt is the analog of M(A) and Ql is the analog of f), see 0.1.2-Q.l. 4.
These two types of splittings commute and consequently Vc, V[ admit Hodge decay, and V.x, V.x• admit Hodge-Tate decay. Theorem: If A is an abelian variety of Ribet type, then Tate cycles (on the abelian variety and all its self-products) are generated by divisors, and therefore the Tate, Hodge, and Mumford-Tate conjectures hold. In this case, MT holds, as well as the Hodge and Tate conjectures (cf. Theorem 3.3.3).
Therefore, the Hodge cycles on the "square" of the abelian 4-fold constructed by Mumford (3.3, Remark) are not all divisorial. So if there is any doubt about the Hodge conjecture (and thus the Tate conjecture), the Weil abelian varieties are the ones to look at. Schoen ([Sc), see also [vG]) succeeded in proving the Hodge conjecture for a family of Weil 4-folds by admitting an action of Q(J.L3.
If A is any 4-dimensional Abelian manifold, then the rings of the Tate cycles and the Hodge cycles coincide (so the Hodge and the Tate conjectures for this manifold are equivalent). Recall (0.2.3), that MT implies that the Hodge and the Tate conjectures are equivalent for an Abelian manifold and all its eigenproducts.
Abelian varieties with reduction conditions
Remark: If N is given by the above matrix, then r = N -129 = logarithm of monodromy, and the sequence in 6.1 corresponds to the filter weight-. Furthermore, if D = Q, then all Hodge and Tate cycles are divisible, so the Hodge and Tate conjectures hold. The importance of abelian varieties of Weil type is not limited to the fact that they (can) have non-dividing Weil cycles (cf. 4.4).
Proof of the algebraicity of the Weil cycles is a critical ingredient in the proof of the Tate conjecture (cf. Theorem: If A is a simple abelian 4-fold with D = Q, admitting bad but not pure multiplicative reduction not, then all the Hodge and the Tate cycles are divisorial, hence the Hodge conjecture, the Tate conjecture and MT hold Proof: The possible values of the toric rank in this case correspond to the pure multiplicative reduction).
Mustafin [Mus] considered abelian varieties with pure multiplicative reduction, but in a geometric setting (i.e. with the operation of the geometric monodromy). The idea of using special element(s) in the representation of the Hodge group has been used before. However, the impotences he considered were of the maximum possible rank, that is, with only one Jordaan block ([Ka 1], ch. 7).
In this case, g, ~ (notation as in 7.1.3) are simple and contain nilpotents r primed for the dimension of the representations. It is reductive and contains the Hodge group (it "sits between" the Hodge and the Lefschetz groups if an abelian variety is not of type III, since V. The Hodge (respectively the Tate) conjecture follows from the equality of the Hodge (respectively the Galois) group to the motivic Galois.
If the (semisimple parts of) Hodge and Galois groups are simple, then we can conclude that either MT holds or the semisimple parts of the Hodge and motivating Galois groups coincide. So, the Hodge conjecture in that case is equivalent to the equality of the centers of these two groups. The consideration of the motivating Galois also implies (as in the proof of Theorem 10.1) that for type III abelian varieties with simple Hodge and Galois groups either MT or the Hodge conjecture holds.
Moduli spaces and reductions
Therefore, we can find a finite set of primes of the ring of integers OF of F such that A;M admits an integral model over 0 := OF[S-1], where S is the product of these primes. Then the fiber Ax over x is isomorphic to the identity component of the Nero model Ax (cf. [BLR], 7.4.3). Indeed, the fibers Ax, Ay are naturally isomorphic to the identity components of Nero.
Xp = Yp exactly means that the special fibers of the identifying components of the Neron models are the same, i.e., Ax, Ar have the same reductions modulo p. Remark: Since Ax has stable reduction, the identity component of the Neron model behaves well with respect to change of basis (cf. [BLR], §7). We want to show that the set of abelian varieties defined over number fields (containing K) that have the same reduction as x is "large".
We are interested in the algebraic points of the modulus space which are p-adically close to x, i.e. have the same reduction mode p. This is a direct proof of the opening of a finite map in the adic topology. In particular, the set of abelian varieties that have bad reductions with fixed toric rank is dense.
First, note that by Poincare's reducibility theorem and Schur's lemma, if A has an abelian variety End0(A) = k, then A is simple. We can assume that the abelian varieties from the analytic subset of the module space discussed in the proof of Theorem 14.1 have exceptional Hodge classes (cf. [W]). For the first density theorem, instead of a (compactification) of the module space, we could consider any family of abelian varieties over a basis of finite type over a number field.
If there is a C-open subset of the base, over which the fibers are simple, then the proof of (part of the) second density theorem (concerning the density of simple manifolds) also continues. One can apply the previous remark to prove the density of simple abelian varieties with a prescribed reduction, regardless of the endomorphisms. If there are simple abelian varieties in the family (see 2. above) this is certainly possible.
Namely, use [Bor] (cf. also [D 1], Proposition 5.2) to show that the set of abelian varieties that are isogenic givens is dense in the corresponding Simur variety. So, for any prime, "most" weakly reduced abelian variants have a minimal reduction at that prime.
