Improved bounds for Serre’s open image theorem

Let $E$ be an elliptic curve over $\mathbb{Q}$ without complex multiplication. Serre showed in [14] that the adelic representation attached to $E/\mathbb{Q}$ has open image, in particular, there is a minimal natural number $C_{E}$ such that the mod $\ell$ representation ${\bar{\rho}}_{E,\ell}$ is surjective for any prime $\ell>C_{E}$.

Based on the method used, the constant $C_{1}$ here is unfortunately rather large. Recent work of Mayle-Wang [9] has given an explicit result on the smallest prime which achieves $a_{p}(E)\neq a_{p}(E^{\prime})$. The constants are quite small, and like Serre, depend on only knowledge of the primes of bad reduction of the two elliptic curves $E$ and $E^{\prime}$.

The method is based on using effective forms of the Chebotarev density theorem together with the Faltings-Serre method, in particular, by studying the ‘deviation group’ $\delta(G)$ of the $2$-adic representations attached to two elliptic curves.

This version requires the results of Rouse and Zureick-Brown [12] which characterizes the images of the $2$-adic representations attached to an elliptic curve over $\mathbb{Q}$.

The Magma computational algebra system [2] was used for verifying assertions in this paper. The electronic resources are available from

https://github.com/ichensfuca/ChenSwidinsky.

Effective versions of Chebotarev’s Density Theorem exist as well and we shall be applying results in which the constants are explicitly computable in terms of the discriminants of $L$ and $K$, as well as the degree of each extension over $\mathbb{Q}$. The first of these was given by Lagarias and Odlyzko [8]. Their result relies on the validity of GRH.

A corollary to the above is given in Mayle-Wang [9, Corollary 6] when we need to pick the prime $p$ to be coprime to a given positive integer $m$.

For a fixed $n_{K}$, we say a triple $(a,b,c)$ is bigger than a triple $(a^{\prime},b^{\prime},c^{\prime})$ (resp. a triple $(a^{\prime},b^{\prime},c^{\prime})$ is smaller than a triple $(a,b,c)$) if

for all values of $\log|d_{K}|$ in a row of [1, Table 3].

We give our own version of Theorem 2.5 and Corollary 2.6. The idea is to collapse [1, Table 3] into a 1-dimensional table, removing the condition on $\log|d_{\tilde{K}}|$ so that each triple is valid for a range of $n_{\tilde{K}}$. We do this by picking a “pivot” triple for each column, for which all triples appearing before the pivot are absorbed into a special constant $p_{0}(n_{\tilde{K}})$, and all triples appearing after are checked to be smaller than the pivot triple.

The pivot triple in each column of [1, Table 3] is chosen to be the first one so that (2.3) holds.

In this section, we wish to construct a finite group, called the deviation group, denoted $\delta(G)$, from which we can find a finite subset that will determine if the two representations are isomorphic or not.

Our treatment of the deviation group will follow the exposition given in Ignasi’s thesis [11]. We note that Ignasi’s exposition is, itself, taken from Chênevert’s thesis [5], whose work follows the work of Serre [16] (the propositions and lemmas which appear here, with the exception of Lemma 3.8, can also be found in [5, Chapter 5]).

Let $G$ be a group, and $L$ be a finite extension of $\mathbb{Q}_{\ell}$, for $\ell$ prime, with ring of integers $\mathcal{O}_{\lambda}$, maximal ideal $\lambda$, and residue field $k=\mathcal{O}_{\lambda}/\lambda\mathcal{O}_{\lambda}$. We let $\pi$ be a uniformizer, so $\lambda=\pi\mathcal{O}_{\lambda}$. Let $\rho_{1},\rho_{2}:G\to\operatorname{GL}_{n}(\mathcal{O}_{\lambda})$ be two $\lambda$-adic representations. We begin by extending the map $\rho_{1}\times\rho_{2}:G\to\operatorname{GL}_{n}(\mathcal{O}_{\lambda})\times\operatorname{GL}_{n}(\mathcal{O}_{\lambda})$ from $G$ to the group ring $\mathcal{O}_{\lambda}[G]$.

We define the map $\rho:\mathcal{O}_{\lambda}[G]\to M_{n}(\mathcal{O}_{\lambda})\oplus M_{n}(\mathcal{O}_{\lambda})$ to be

Before we introduce the next proposition, some further explanations are needed (following [11]). We assume now the representations $\rho_{1},\rho_{2}:G\to\operatorname{GL}_{n}(\mathcal{O}_{\lambda})$ are not isomorphic, that is, they are not conjugate in $\operatorname{GL}_{n}(\mathcal{O}_{\lambda})$, but that the residual representations $\bar{\rho}_{1}$ and $\bar{\rho}_{2}$ obtained from $\rho_{1}$ and $\rho_{2}$ by reduction modulo $\lambda$ are isomorphic. We then have an equality $\bar{\rho}_{1}=P\bar{\rho}_{2}P^{-1}$ for some matrix $P\in M_{n}(k)$.

Define $\beta$ to be the largest integer such that $\rho_{1}$ and $\rho_{2}$ are conjugated modulo $\lambda^{\beta}$, that is, there is a matrix $P\in\operatorname{GL}_{n}(\mathcal{O}_{\lambda})$ such that $\rho_{1}\equiv P\rho_{2}P^{-1}\ (\mathrm{mod}\ \lambda^{\beta})$; we then have $\beta\geq 1$, since $\bar{\rho}_{1}\cong\bar{\rho}_{2}$. In addition, there is an integer $\alpha\geq 1$ such that $\operatorname{tr}(\rho_{1})\equiv\operatorname{tr}(\rho_{2})\ (\mathrm{mod}\ \lambda^{\alpha})$ and $\operatorname{tr}(\rho_{1})\not\equiv\operatorname{tr}(\rho_{2})\ (\mathrm{mod}\ \lambda^{\alpha+1})$; in particular, $\rho_{1}$ and $\rho_{2}$ are not conjugate modulo $\lambda^{\alpha+1}$, so $\beta\leq\alpha$. Given that $\rho_{1}$ and $\rho_{2}$ are conjugate modulo $\lambda^{\beta}$ but not conjugate modulo $\lambda^{\beta+1}$, if we replace $\rho_{2}$ with a conjugate we may assume $\rho_{1}\equiv\rho_{2}\ (\mathrm{mod}\ \lambda^{\beta})$ but $\rho_{1}\not\equiv\rho_{2}\ (\mathrm{mod}\ \lambda^{\beta+1})$.

Hence, for any $g\in G$, we have

for some $\theta_{g}\in M_{n}(\mathcal{O}_{\lambda})$ and $\pi$ a uniformizer of $\lambda$. Rearranging, we get an equation for $\rho_{2}(g)$ of the form

where $I_{n}$ is the $n\times n$ identity matrix. Let $\theta:G\to M_{n}(\mathcal{O}_{\lambda})$ be the map $g\to\theta_{g}\rho_{1}(g)^{-1}$, and notice that (3.2) becomes

It can be difficult to compute the exact size of $\delta(G)$, or find a tighter upper bound for it. We will, in the following section, work to replace $\delta(G)$ with $\varphi(G)$ in the case of $2$-adic representations. The codomain of $\varphi$ is easily understood, and hence a bound for $|\varphi(G)|$ is easily computable. This is what allows us to prove Theorem 1.3.

We now give an analogous version of Proposition 4.1 in the case where the mod 2 representations are isomorphic and absolutely irreducible. This allows us to replace $\delta(G)$ in Proposition 4.1 with $\varphi(G)$ from Proposition 3.6, a set which is easier to estimate the size of. The idea to replace $\delta(G)$ with $\varphi(G)$ comes from Chênevert [5, pg. 114], in which he gives a remark that, in the $2$-adic case, Serre [16] implies that $\delta(G)\cong\varphi(G)$. However, in a conversation with Chênevert, Serre mentions he might not have proven the map $\delta(G)\to\varphi(G)$ was an isomorphism, but, in an unpublished letter to Tate, that the $\alpha$ in the proof of Proposition 3.5 is equal to the $\beta$ coming from the construction of the function $\varphi$ if the residual representation is surjective. We show, in the 2-adic case, that $\alpha=\beta$, and that we can replace $\delta(G)$ in Proposition 4.1 with $\varphi(G)$ and get the same conclusion, that is, there is a subset $C\subseteq\varphi(G)$ that is a conjugacy class, and if $g\in G$ is such that $\varphi(g)\in C$, then $\operatorname{tr}\rho_{1}(g)\neq\operatorname{tr}\rho_{2}(g)$.