A local-to-global principle for isogeny

Question

If two elliptic curves over $\mathbb{Q}$ are $\mathbb{Q}_p$-isogneous for almost all primes $p$, then they are $\mathbb{Q}$-isogenous.

This follows from the fact that they have the same number of $\mathbb{F}_p$-points for almost all $p$, hence their $L$-functions have the same local factors at all these $p$, therefore a combination of "multiplicity one" and Faltings' isogeny theorem implies that they are $\mathbb{Q}$-isogenous. Correct me if I'm wrong.

Here $\mathbb{Q}$ can be replaced by any number field $k$.

Question : Does the same argument work for any two abelian varieties $A$, $B$ over $k$ ? It should, since $H^i$ is $\wedge^i H^1$ for $i>0$.

If so, this explains why Poonen's abelian surfaces $A,B$ (everywhere locally isomorphic but not isomorphic) are $\mathbb{Q}$-isogenous.

Answer 1

Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem [and, as Kevin Buzzard points out, of the semisimplicity of the Galois action on the Tate module, also proved by Faltings.]

The proof for abelian varieties is almost the same as that for elliptic curves. You just need to observe that if A, A' are two g-dimensional abelian varieties over F_q, then the following are equivalent:

(i) They are $F_q$-rationally isogenous.
(ii) They have the same characteristic polynomial of Frobenius.
(iii) For all $1 \leq i \leq g, \ |A(F_{q^i})| = |A'(F_{q^i})|$.
(iv) The Hasse-Weil zeta functions of A and A' coincide.

Although I have not checked in order to answer this question, I think it is likely that proofs -- or references to proofs -- of this fact can be found in at least one of the papers

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Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521--560.

Waterhouse, W. C.; Milne, J. S. Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53--64. Amer. Math. Soc., Providence, R.I., 1971.

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Answer 2

Let me add a different perspective on Stefan Keil's question: why is the isogeny class of an elliptic curve $E/\mathbb{Q}$ determined by the sequence of numbers $|E(\mathbb{F}_p)|$ for (almost) all primes $p$? This of course is a particular case of Faltings' isogeny theorem (the $L$-function determines the isogeny class), but it is possible to proceed differently, by a more elementary route in diophantine approximations.

The conclusion of the isogeny theorem is an algebraicity statement; its assumptions are arithmetic. There is a long string of results of such flavor in the literature, starting with the discovery of E. Borel that a meromorphic function on a disc of radius $R$ is rational (an algebraicity conclusion) if $R > 1$ and its Taylor expansion at the center of the disc has $\mathbb{Z}$-coefficients (an arithmeticity assumption). The general shape of such "diophantine algebraicity" criteria is the following:

If a formal arithmetical object (for example, a power series in $\mathbb{Z}[[T]]$, or a formal subscheme of an arithmetic scheme) admits a complex uniformization by meromorphic functions of finite order living on a sufficiently large region (an example: in the context of elliptic curves and their associated objects, consider the Weierstrass $\wp$ function, which is meromorphic of order $2$ on all of $\mathbb{C}$), then the formal object is algebraic.

Ultimately, such statements boil down to the product formula. Very roughly speaking, the complex uniformization yields a growth condition on "the coefficients" of a formal object, which collides with the integrality assumption to yield that almost all "coefficients" vanish, thereby yielding the desired algebraic relation.

D.V. and G.V. Chudnovsky have a sequence of papers from the 1980s in which they obtain such algebraicity and algebraic dependency criteria, and apply them to problems such as the Grothendieck-Katz $p$-curvature conjecture. In Padé approximations and diophantine geometry [Proc. Natl. Acad. Sci. USA, 1985], using a result of Honda on formal group laws over $\mathbb{Z}$, they deduce the statement of the opening paragraph (the isogeny theorem for elliptic curves over $\mathbb{Q}$) from just such a diophantine criterion for algebraic dependency of two power series in $\mathbb{Z}[[T]]$. Their work was extended by Graftieaux (Formal groups and the isogeny theorem) to abelian varieties over $\mathbb{Q}$ with real multiplication.

Chudnovsky's algebraicity criteria were vastly generalized by J.-B. Bost to leaves of algebraic foliations. As an application of his general algebraicity criterion, Bost proved the following:

Theorem. (J.-B. Bost) Let $G$ be an algebraic group over $\mathbb{Q}$. Consider a Lie subalgebra $\mathfrak{h} \subset \mathrm{Lie}(G)$ whose reductions $\mathfrak{h}\mod{p}$ at almost all primes $p$ are stable under $p$-th powers. Then there is an algebraic subgroup $H$ of $G$ such that $\mathfrak{h} = \mathrm{Lie}(H)$.

The statement of the opening paragraph is an easy consequence. Let $E,E'$ be elliptic curves over $\mathbb{Q}$ having $|E(\mathbb{F}_p)| = |E'(\mathbb{F}_p)|$ for almost all $p$. Apply Bost's theorem with $G:= E \times E'$ and $\mathfrak{h}$ an arbitrary one-dimensional horizontal $\mathbb{Q}$-vector subspace of $\mathrm{Lie}(G) = \mathrm{Lie}(E) \oplus \mathrm{Lie}(E')$. The $p$-th power operation on $\mathrm{Lie}(E_p)$ (resp. $\mathrm{Lie}(E'_p)$) is induced by Frobenius, and is equal (mod $p$) to multiplication by $1 -|E(\mathbb{F}_p)|$ (resp., by $1-|E'(\mathbb{F}_p)|$). If $|E(\mathbb{F}_p)| = |E'(\mathbb{F}_p)|$ then $\mathfrak{h}\mod{p}$ is stable under the $p$-th power operation, and the conclusion follows from Bost's theorem.

Similarly, the consideration of $G := \mathbb{G}_m \times \mathbb{G}_m$ leads as another consequence to a diophantine approximations proof of the special case of the Frobenius-Chebotarev theorem according to which a finite extension of number fields is trivial if almost all primes split completely. Let me recall that in the proof of global class-field theory, this is the key point behind the "Second Main inequality" supplying enough norms from every finite extension $L/K$ of a number field; it is the only place where the traditional proofs had to resort to analytic arguments involving the $L$-function. With his introduction of the ideles, Chevalley took great pains to give a purely algebraic proof in his classic 1940 Annals paper. Here we have a diophantine approximations proof.

For more applications of diophantine algebraicity criteria, see A. Chambert-Loir's Bourbaki expose (Théorèmes d'algebricité en géométrie diophantienne).

Answer 3

A propos Dimitrov's answer, one should also mention two beautiful papers of Masser and Wustholz.

MR1037140 Masser, D. W.(1-MI); Wüstholz, G.(CH-ETHZ); Estimating isogenies on elliptic curves. Invent. Math. 100 (1990), no. 1, 1–24.

MR1217345 Masser, David(1-MI); Wüstholz, Gisbert(CH-ETHZ); Isogeny estimates for abelian varieties, and finiteness theorems. Ann. of Math. (2) 137 (1993), no. 3, 459–472.

Given a number field $K$ and abelian varieies $A/K$ and $A'/K$ that are isogenous over $K$, they give an explicit value $M(K,A,A')$ such that there is a $K$-isogeny $A\to A'$ of degree less then $M$. (The first article treats the case of elliptic curves.) Such estimates have many applications, including a proof of the hardest implication of the theorem stated in Keller's answer (by an argument appearing in Tate's original paper). As in the articles by the Chudnovskys, the tools used in the proof are those of Diophantine approxmation and transcendence theory.

Answer 4

@Stefan Keil: See Faltings' Finiteness Theorems for Abelian Varieties over Number Fields, Corollary 2:

For Abelian varieties $A$, $B$ over a number field $K$, the following are equivalent:

• $A$ and $B$ are isogenous

• $T_\ell A \otimes \mathbf{Q}_\ell \cong T_\ell B \otimes \mathbf{Q}_\ell$ as $\pi$-modules

• $L_v(A,s) = L_v(B,s)$ for almost all places $v$ of $K$.

• $L_v(A,s) = L_v(B,s)$ for all places $v$ of $K$.