--- title: Effective Open Image Theorem url: https://www.emergentmind.com/topics/effective-open-image-theorem type: topic --- # Effective Open Image Theorem 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 [2212.11472]. ## 1. Geometric and representation-theoretic setting For a principally polarized abelian variety $B/K$ of dimension $g$, the $\ell$-adic Tate module $$ T_\ell(B)\simeq \mathbb Z_\ell^{2g} $$ carries a Galois representation $$ \rho_{B,\ell}:\operatorname{Gal}(\overline K/K)\to \operatorname{GSp}_{2g}(\mathbb Z_\ell), $$ whose multiplier is the $\ell$-adic cyclotomic character [2212.11472]. 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 $$ A=A_1\times\cdots\times A_n, $$ the image is not expected to fill an unrestricted product $\prod_i \operatorname{GSp}_{2g}(\mathbb Z_\ell)$, because all factors have the same cyclotomic multiplier. The natural target is therefore the diagonal similitude subgroup $$ \Delta_{2g,n}(\widehat{\mathbb Z}) := \left\{ (\gamma_1,\dots,\gamma_n)\in \prod_{i=1}^n \operatorname{GSp}_{2g}(\widehat{\mathbb Z}) :\operatorname{mult}\gamma_1=\cdots=\operatorname{mult}\gamma_n \right\}. $$ In the corresponding $\ell$-adic setting one replaces $\widehat{\mathbb Z}$ by $\mathbb Z_\ell$. 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 $\Delta$-group determined by the common multiplier constraint [2212.11472]. 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 $\Delta_{2g,n}(\widehat{\mathbb Z})$ and to the existence of a finite surjectivity threshold $c(A)$ [2212.11472]. ## 2. The constant $c(A)$ and the main explicit bound The threshold is defined by $$ c(A):=\min\{C>0:\forall \text{ primes }\ell>C,\ \rho_{A,\ell}(\operatorname{Gal}(\overline K/K))=\Delta_{2g,n}(\mathbb Z_\ell)\}. $$ Thus $c(A)$ is the smallest integer such that every sufficiently large prime $\ell$ yields full $\ell$-adic image in the maximal permitted group [2212.11472]. To state the explicit bound, one introduces $\ell_g$, the smallest prime not dividing $2g$, and the auxiliary quantities $$ c(K,g)=2(\ell_g^{8g^2}-1)^2\Bigl[\log d_K+[K:\mathbb Q]\log\bigl(2(\ell_g^{8g^2}-1)^2\bigr)\Bigr], $$ and $$ c(K,A_i,A_j)=2(\ell_g^{8g^2}-1)^2[K:\mathbb Q]\log \operatorname{rad}(2\ell_gN_{A_i}N_{A_j}d_K), $$ where $d_K=|\operatorname{Disc}K|$, $N_{A_i}$ is the conductor norm of $A_i$, and $\operatorname{rad}(\cdot)$ denotes the radical [2212.11472]. Effective Chebotarev with avoidance supplies absolute constants $\tilde a,\tilde b,\tilde c$; under GRH one may take $$ \tilde a=4,\qquad \tilde b=2.5,\qquad \tilde c=5. $$ With this notation, the theorem asserts that the following are equivalent: 1. for all $i\neq j$, the varieties $A_i$ and $A_j$ are not isogenous over $\overline K$; 2. the adelic image $G_A$ is an open subgroup of $\Delta_{2g,n}(\widehat{\mathbb Z})$; 3. the integer $c(A)$ exists and satisfies $$ c(A_1\times\cdots\times A_n)\le \max_{1\le iC$ is equivalent to checking it simultaneously on each pair $(i,j)$, so the proof reduces to studying $A_i\times A_j$ [2212.11472]. This reduction is the mechanism by which a product bound is expressed through the maxima over pairs in the explicit formula for $c(A)$. ## 4. Proof architecture and key intermediate results The proof combines group theory, effective isogeny theory, and effective Chebotarev. One central input is a classification of subgroups $$ G\subset \Delta_{2g}(\mathbb F_\ell) $$ that project surjectively to both factors. Proposition 3.11 shows that such a subgroup is one of three types: the graph of an automorphism in $\operatorname{Aut}_{\mathrm{mult}}(\operatorname{GSp}_{2g}(\mathbb F_\ell))$, a projective graph via $\operatorname{Aut}_{\mathrm{mult}}(\operatorname{GSp}/\{\pm I\})$, or the full group $\Delta_{2g}(\mathbb F_\ell)$. From this one deduces a trace criterion: by Corollary 3.12, one has $G=\Delta_{2g}(\mathbb F_\ell)$ as soon as there exists $(\gamma_1,\gamma_2)\in G$ with $$ \operatorname{tr}\gamma_1\notin\{\pm \operatorname{tr}\gamma_2\} $$ [2212.11472]. A second input is the lifting step from mod $\ell$ to $\mathbb Z_\ell$. Proposition 2.7 states that if the mod-$\ell$ image contains $\operatorname{Sp}\times \operatorname{Sp}$, then the $\ell$-adic image contains $\operatorname{Sp}\times \operatorname{Sp}$ over $\mathbb Z_\ell$. This converts finite-field maximality into full $\ell$-adic maximality [2212.11472]. The arithmetic input is an effective Faltings isogeny theorem. Under GRH, if $A_1$ and $A_2$ are nonisogenous of the same dimension $g$ and each has open adelic image, Theorem 4.3 furnishes a good-reduction prime $\mathfrak p$ with $$ N(\mathfrak p)\le \Bigl[\tilde a\bigl(c(K,g)+c(K,A_1,A_2)\bigr)+2\tilde b(\ell_g^{8g^2}-1)^2+\tilde c\Bigr]^2 $$ such that $$ a_{\mathfrak p}(A_1)\neq \pm a_{\mathfrak p}(A_2). $$ This separating prime is the key device for forcing trace discrepancy in the mod-$\ell$ image [2212.11472]. The proof then proceeds as follows. First, the product problem is reduced to the pairwise case. Next, for a fixed pair $A_1,A_2$, the effective isogeny theorem produces a prime $\mathfrak p$ at which the Frobenius traces differ up to sign. Writing $$ B:= \Bigl[\tilde a\bigl(c(K,g)+c(K,A_1,A_2)\bigr)+2\tilde b(\ell_g^{8g^2}-1)^2+\tilde c\Bigr]^2, $$ Weil’s bound shows that any $\ell>4g\sqrt B$ cannot divide $a_{\mathfrak p}(A_1)\pm a_{\mathfrak p}(A_2)$ [2212.11472]. If one also assumes $$ \ell>\max\{4g\sqrt B,c(A_1),c(A_2)\}, $$ then the individual mod-$\ell$ images are full $\operatorname{GSp}_{2g}(\mathbb F_\ell)$ by definition of $c(A_i)$, the trace criterion forces the product mod-$\ell$ image to be $\Delta_{2g}(\mathbb F_\ell)$, and Proposition 2.7 lifts this to $$ G_{A_1\times A_2,\ell}=\Delta_{2g}(\mathbb Z_\ell). $$ Finally, combining all pairs yields the stated bound for $A_1\times\cdots\times A_n$, while openness of the adelic image follows because only finitely many small $\ell$ remain to be checked [2212.11472]. ## 5. GRH, effectivity, and the shape of the constants In the supplied formulation, the only step requiring GRH is the effective Chebotarev theorem used to produce a small prime with prescribed Frobenius and with avoidance conditions. Corollary 2.12 gives, under GRH, an unramified prime $\mathfrak p$ in a prescribed union of Frobenius classes satisfying $$ N(\mathfrak p)\le [\tilde a\log d_{\widetilde L}+\tilde b[L:K]+\tilde c]^2, $$ with effective control of $\log d_{\widetilde L}$ in terms of $d_K$, $[K:\mathbb Q]$, conductors, and related invariants [2212.11472]. This is exactly what converts a qualitative open image theorem into an explicit upper bound for $c(A)$. The theorem is therefore **conditional effective** rather than unconditional in the strongest quantitative sense. Under GRH one may take $(\tilde a,\tilde b,\tilde c)=(4,2.5,5)$, and these constants propagate directly into the bound for $c(A)$ [2212.11472]. Without GRH, the discussion states that one still obtains an explicit bound via unconditional effective Chebotarev, but it is exponentially larger [2212.11472]. The shape of the final estimate reflects three arithmetic inputs. The field $K$ contributes through $d_K$ and $[K:\mathbb Q]$. The pairwise interaction of factors contributes through the conductor norms $N_{A_i}$ and $N_{A_j}$. The individual factors contribute through the prior constants $c(A_i)$, which, in the surrounding literature, are controlled using known bounds involving invariants such as Faltings height. This suggests that the theorem is best viewed as a synthesis: pairwise nonisogeny is detected by Frobenius trace separation, while individual largeness of image is imported from single-factor effective open image results. ## 6. Relation to other effective open image theorems The effective open image problem originates in Serre’s theorem for elliptic curves, where one asks for explicit control of the largest prime $\ell$ for which the mod-$\ell$ representation can fail to be surjective. For a non-CM elliptic curve $E/\mathbb Q$, Mayle and Wang proved under GRH that $$ c(E)\le 964\cdot \log \operatorname{rad}(2N_E)+5760 $$ [2109.08656]. Chen and Swidinsky subsequently improved this to $$ C_E\le 446\cdot \log \operatorname{rad}(2N_E)+2254 $$ under GRH [2501.00202]. These results concern single elliptic curves over $\mathbb Q$ and furnish explicit linear-in-log-conductor bounds. For products of elliptic curves over a number field, Lombardo gave an explicit adelic index bound for the image of $$ \rho_\infty:\operatorname{Gal}(\overline K/K)\to \prod_{i=1}^n \operatorname{GL}_2(\widehat{\mathbb Z}) $$ attached to pairwise nonisogenous non-CM curves $E_1,\dots,E_n/K$. In that setting the image is open in $$ \Delta=\{(g_1,\dots,g_n)\in \operatorname{GL}_2(\widehat{\mathbb Z})^n:\det g_1=\cdots=\det g_n\}, $$ and the index is bounded explicitly in terms of $[K:\mathbb Q]$ and isogeny-theoretic quantities built from Faltings heights [1501.04600]. The emphasis there is on adelic index bounds, whereas the later product theorem for principally polarized abelian varieties isolates an explicit eventual surjectivity threshold $c(A)$ [2212.11472]. A broader generalization is Zywina’s effective open image theorem for abelian varieties. For an abelian variety $A/K$ of dimension $g$, he proved that for $\ell$ larger than a computable threshold of the shape $$ \ell\ge c\max\{[K:\mathbb Q],h(A),N(\mathfrak q)\}, $$ with constants depending only on $g$, the index $$ [G_{A,\ell}(\mathbb Z_\ell):\rho_{A,\ell}(\operatorname{Gal}_K)] $$ is bounded in terms of $g$ and, in the unramified connected case, by a constant depending only on $g$ [1910.14171]. That theorem controls the index inside the $\ell$-adic monodromy group for large $\ell$. 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 [2212.11472]. 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-$\ell$ or $\ell$-adic surjectivity, and large-$\ell$ control of indices inside monodromy groups. In the product setting for principally polarized abelian varieties, the theorem of [2212.11472] is notable for combining these themes into a pairwise criterion with a completely explicit bound for $c(A)$ under GRH. Source: https://www.emergentmind.com/topics/effective-open-image-theorem