Effective Open Image Theorem

• The paper establishes an explicit GRH-conditional bound c(A) such that for all primes ℓ exceeding c(A), the ℓ-adic representation surjects onto the specified diagonal similitude group.
• It employs effective Chebotarev density, isogeny theory, and Frobenius trace separation to control the adelic image of products of nonisogenous, principally polarized abelian varieties.
• The refinement quantifies exceptional primes by linking the maximal open image condition to concrete invariants like conductor norms and the discriminant of the base field.

Searching arXiv for related work on effective open image theorems and the specific 2022 paper. The Effective Open Image Theorem refers to an explicit form of an open image statement for Galois representations attached to abelian varieties, in which one does not merely prove that the image is open of finite index, but also gives a computable threshold beyond which the local $\ell$-adic images are exactly the largest groups permitted by the polarization and cyclotomic multiplier constraints. In the formulation made explicit for products of principally polarized abelian varieties, one considers

$A=A_1\times\cdots\times A_n$

over a number field $K$, with $n\ge 2$, all factors of common dimension $g$, each principally polarized, each with trivial geometric endomorphism ring, and pairwise nonisogenous over $\overline K$. Under these hypotheses, and assuming the Generalized Riemann Hypothesis, one obtains an explicit upper bound for the minimal constant $c(A)$ such that for every prime $\ell>c(A)$ the $\ell$-adic Galois representation of $A$ is surjective onto the appropriate diagonal similitude group; equivalently, the full adelic image is open in that group (Mayle et al., 2022).

1. Geometric and representation-theoretic setting

For a principally polarized abelian variety $A=A_1\times\cdots\times A_n$0 of dimension $A=A_1\times\cdots\times A_n$1, the $A=A_1\times\cdots\times A_n$2-adic Tate module

$A=A_1\times\cdots\times A_n$3

carries a Galois representation

$A=A_1\times\cdots\times A_n$4

whose multiplier is the $A=A_1\times\cdots\times A_n$5-adic cyclotomic character (Mayle et al., 2022). The symplectic similitude group appears because the principal polarization supplies a Weil pairing, and the multiplier records the Galois action on roots of unity.

For a product

$A=A_1\times\cdots\times A_n$6

the image is not expected to fill an unrestricted product $A=A_1\times\cdots\times A_n$7, because all factors have the same cyclotomic multiplier. The natural target is therefore the diagonal similitude subgroup

$A=A_1\times\cdots\times A_n$8

In the corresponding $A=A_1\times\cdots\times A_n$9-adic setting one replaces $K$0 by $K$1. A common misconception is that “maximal image” should mean the full direct product of symplectic similitude groups; in this context, the largest possible image is instead the relevant $K$2-group determined by the common multiplier constraint (Mayle et al., 2022).

The effective theorem is an explicit refinement of an open image theorem of Hindry and Ratazzi. In the self-contained statement under discussion, the pairwise nonisogeny condition on the factors is not merely technical: it is equivalent to the adelic openness of the product image in $K$3 and to the existence of a finite surjectivity threshold $K$4 (Mayle et al., 2022).

2. The constant $K$5 and the main explicit bound

The threshold is defined by

$K$6

Thus $K$7 is the smallest integer such that every sufficiently large prime $K$8 yields full $K$9-adic image in the maximal permitted group (Mayle et al., 2022).

To state the explicit bound, one introduces $n\ge 2$0, the smallest prime not dividing $n\ge 2$1, and the auxiliary quantities

$n\ge 2$2

and

$n\ge 2$3

where $n\ge 2$4, $n\ge 2$5 is the conductor norm of $n\ge 2$6, and $n\ge 2$7 denotes the radical (Mayle et al., 2022).

Effective Chebotarev with avoidance supplies absolute constants $n\ge 2$8; under GRH one may take

$n\ge 2$9

With this notation, the theorem asserts that the following are equivalent:

1. for all $g$0, the varieties $g$1 and $g$2 are not isogenous over $g$3;
2. the adelic image $g$4 is an open subgroup of $g$5;
3. the integer $g$6 exists and satisfies

$g$7

This gives an explicit GRH-conditional upper bound for the largest exceptional prime in terms of standard invariants of $g$8 and the factors $g$9 (Mayle et al., 2022).

The dependence on the individual constants $\overline K$0 shows that the product theorem is not autonomous: it builds on prior control of the single-factor open image problem. The supplied discussion further notes that Faltings heights enter through known bounds on $\overline K$1 (Mayle et al., 2022).

3. Structural meaning of “open image” and “largest possible image”

The theorem distinguishes two related but nonidentical assertions. The first is adelic openness, meaning that

$\overline K$2

is open, hence of finite index. The second is eventual local surjectivity, meaning that for all sufficiently large primes $\overline K$3 the image of the $\overline K$4-adic representation is exactly

$\overline K$5

The constant $\overline K$6 measures when the second phenomenon begins (Mayle et al., 2022).

In this setting, openness of the adelic image is equivalent to the existence of a finite exceptional set of primes where surjectivity may fail. The theorem makes that equivalence effective. It also shows that the obstruction to maximality is entirely encoded, for large $\overline K$7, by isogenies among the factors: pairwise nonisogeny over $\overline K$8 is exactly the criterion for openness and eventual surjectivity (Mayle et al., 2022).

A further structural reduction is that the $\overline K$9-factor problem is controlled by pairwise products. Lemma 5.1 shows that checking $c(A)$0-surjectivity for all $c(A)$1 is equivalent to checking it simultaneously on each pair $c(A)$2, so the proof reduces to studying $c(A)$3 (Mayle et al., 2022). This reduction is the mechanism by which a product bound is expressed through the maxima over pairs in the explicit formula for $c(A)$4.

4. Proof architecture and key intermediate results

The proof combines group theory, effective isogeny theory, and effective Chebotarev. One central input is a classification of subgroups

$c(A)$5

that project surjectively to both factors. Proposition 3.11 shows that such a subgroup is one of three types: the graph of an automorphism in $c(A)$6, a projective graph via $c(A)$7, or the full group $c(A)$8. From this one deduces a trace criterion: by Corollary 3.12, one has $c(A)$9 as soon as there exists $\ell>c(A)$0 with

$\ell>c(A)$1

(Mayle et al., 2022).

A second input is the lifting step from mod $\ell>c(A)$2 to $\ell>c(A)$3. Proposition 2.7 states that if the mod-$\ell>c(A)$4 image contains $\ell>c(A)$5, then the $\ell>c(A)$6-adic image contains $\ell>c(A)$7 over $\ell>c(A)$8. This converts finite-field maximality into full $\ell>c(A)$9-adic maximality (Mayle et al., 2022).

The arithmetic input is an effective Faltings isogeny theorem. Under GRH, if $\ell$0 and $\ell$1 are nonisogenous of the same dimension $\ell$2 and each has open adelic image, Theorem 4.3 furnishes a good-reduction prime $\ell$3 with

$\ell$4

such that

$\ell$5

This separating prime is the key device for forcing trace discrepancy in the mod-$\ell$6 image (Mayle et al., 2022).

The proof then proceeds as follows. First, the product problem is reduced to the pairwise case. Next, for a fixed pair $\ell$7, the effective isogeny theorem produces a prime $\ell$8 at which the Frobenius traces differ up to sign. Writing

$\ell$9

Weil’s bound shows that any $A$0 cannot divide $A$1 (Mayle et al., 2022). If one also assumes

$A$2

then the individual mod-$A$3 images are full $A$4 by definition of $A$5, the trace criterion forces the product mod-$A$6 image to be $A$7, and Proposition 2.7 lifts this to

$A$8

Finally, combining all pairs yields the stated bound for $A$9, while openness of the adelic image follows because only finitely many small $A=A_1\times\cdots\times A_n$00 remain to be checked (Mayle et al., 2022).

5. GRH, effectivity, and the shape of the constants

In the supplied formulation, the only step requiring GRH is the effective Chebotarev theorem used to produce a small prime with prescribed Frobenius and with avoidance conditions. Corollary 2.12 gives, under GRH, an unramified prime $A=A_1\times\cdots\times A_n$01 in a prescribed union of Frobenius classes satisfying

$A=A_1\times\cdots\times A_n$02

with effective control of $A=A_1\times\cdots\times A_n$03 in terms of $A=A_1\times\cdots\times A_n$04, $A=A_1\times\cdots\times A_n$05, conductors, and related invariants (Mayle et al., 2022). This is exactly what converts a qualitative open image theorem into an explicit upper bound for $A=A_1\times\cdots\times A_n$06.

The theorem is therefore conditional effective rather than unconditional in the strongest quantitative sense. Under GRH one may take $A=A_1\times\cdots\times A_n$07, and these constants propagate directly into the bound for $A=A_1\times\cdots\times A_n$08 (Mayle et al., 2022). Without GRH, the discussion states that one still obtains an explicit bound via unconditional effective Chebotarev, but it is exponentially larger (Mayle et al., 2022).

The shape of the final estimate reflects three arithmetic inputs. The field $A=A_1\times\cdots\times A_n$09 contributes through $A=A_1\times\cdots\times A_n$10 and $A=A_1\times\cdots\times A_n$11. The pairwise interaction of factors contributes through the conductor norms $A=A_1\times\cdots\times A_n$12 and $A=A_1\times\cdots\times A_n$13. The individual factors contribute through the prior constants $A=A_1\times\cdots\times A_n$14, which, in the surrounding literature, are controlled using known bounds involving invariants such as Faltings height. This suggests that the theorem is best viewed as an overview: pairwise nonisogeny is detected by Frobenius trace separation, while individual largeness of image is imported from single-factor effective open image results.

6. Relation to other effective open image theorems

The effective open image problem originates in Serre’s theorem for elliptic curves, where one asks for explicit control of the largest prime $A=A_1\times\cdots\times A_n$15 for which the mod-$A=A_1\times\cdots\times A_n$16 representation can fail to be surjective. For a non-CM elliptic curve $A=A_1\times\cdots\times A_n$17, Mayle and Wang proved under GRH that

$A=A_1\times\cdots\times A_n$18

(Mayle et al., 2021). Chen and Swidinsky subsequently improved this to

$A=A_1\times\cdots\times A_n$19

under GRH (Chen et al., 2024). These results concern single elliptic curves over $A=A_1\times\cdots\times A_n$20 and furnish explicit linear-in-log-conductor bounds.

For products of elliptic curves over a number field, Lombardo gave an explicit adelic index bound for the image of

$A=A_1\times\cdots\times A_n$21

attached to pairwise nonisogenous non-CM curves $A=A_1\times\cdots\times A_n$22. In that setting the image is open in

$A=A_1\times\cdots\times A_n$23

and the index is bounded explicitly in terms of $A=A_1\times\cdots\times A_n$24 and isogeny-theoretic quantities built from Faltings heights (Lombardo, 2015). The emphasis there is on adelic index bounds, whereas the later product theorem for principally polarized abelian varieties isolates an explicit eventual surjectivity threshold $A=A_1\times\cdots\times A_n$25 (Mayle et al., 2022).

A broader generalization is Zywina’s effective open image theorem for abelian varieties. For an abelian variety $A=A_1\times\cdots\times A_n$26 of dimension $A=A_1\times\cdots\times A_n$27, he proved that for $A=A_1\times\cdots\times A_n$28 larger than a computable threshold of the shape

$A=A_1\times\cdots\times A_n$29

with constants depending only on $A=A_1\times\cdots\times A_n$30, the index

$A=A_1\times\cdots\times A_n$31

is bounded in terms of $A=A_1\times\cdots\times A_n$32 and, in the unramified connected case, by a constant depending only on $A=A_1\times\cdots\times A_n$33 (Zywina, 2019). That theorem controls the index inside the $A=A_1\times\cdots\times A_n$34-adic monodromy group for large $A=A_1\times\cdots\times A_n$35. By contrast, the 2022 product theorem treats a setting where the expected maximal group is explicitly identified as a diagonal symplectic similitude group and provides a GRH-conditional closed formula for the threshold after which surjectivity holds at every prime (Mayle et al., 2022).

Taken together, these works show that the phrase “effective open image theorem” covers several distinct quantitative problems: explicit adelic index bounds, explicit bounds for exceptional primes in mod-$A=A_1\times\cdots\times A_n$36 or $A=A_1\times\cdots\times A_n$37-adic surjectivity, and large-$A=A_1\times\cdots\times A_n$38 control of indices inside monodromy groups. In the product setting for principally polarized abelian varieties, the theorem of (Mayle et al., 2022) is notable for combining these themes into a pairwise criterion with a completely explicit bound for $A=A_1\times\cdots\times A_n$39 under GRH.