nt.number theory - A local-to-global principle for isogeny

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).