Main Theorem 1. Let \(E\) be an elliptic curve over \(\mathbb{Q}\). Then, \(\boldsymbol{a}=(a_{p}(E)\bmod p)_{p}\) is \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental.
May 17, 2026
The first purpose of this paper is to give the fnite transcendence of Frobenius traces for elliptic curves over \(\mathbb{Q}\) without the assumption of complex multiplication (CM). This result generalizes the previous work by Luca and Zudilin, who obtained similar transcendence results specifically for the CM case. The second purpose is to give the finite transcendence of Frobenius traces for several principally polarized abelian varieties over \(\mathbb{Q}\), by using Luca–Zudilin’s method.
The finite algebraic number which lives in the ring \[\begin{align} \nonumber \mathcal{A}=\left. \prod_{p:\text{prime}}\mathbb{F}_{p}\middle/\bigoplus_{p:\text{prime}}\mathbb{F}_{p} \right. \end{align}\] was originally introduced by Rosen ([1]) in 2020. Rosen’s paper consisted of constructing analogues of \(\overline{\mathbb{Q}}\) and (complex) periods within \(\mathcal{A}\) and investigating their properties (e.g. algebraic structure). The set of finite algebraic numbers defined in [1] is written as \(\mathcal{P}_{\mathcal{A}}^{0}\). It can be characterized as follows:
Theorem 1 ([1]). The followings are equivalent:
An element \(\boldsymbol{t} = (t_{p} \bmod p)_{p} \in \mathcal{P}_{\mathcal{A}}^{0}\).
There exists a linear recurrent sequence \((a_{n})_{n}\in\mathbb{Q}^{\mathbb{N}}\) such that \(\boldsymbol{t}=(a_{p}\bmod p)_{p}\).
There exists a finite Galois extension \(L/\mathbb{Q}\) and a map \(g \colon \mathrm{Gal}(L/\mathbb{Q}) \to L\) satisfying \[\begin{align} \nonumber g(\sigma\tau\sigma^{-1}) = \sigma g(\tau) \quad \text{for all } \sigma, \tau \in \mathrm{Gal}(L/\mathbb{Q}), \end{align}\] such that \[\begin{align} \nonumber \boldsymbol{t} = \left( g(\mathrm{Frob}_{\mathfrak{P}}) \bmod \mathfrak{P}\right)_{p}. \end{align}\] Here, \(\mathfrak{P}\mid p\) is a prime ideal in \(L\) and \(\mathrm{Frob}_{\mathfrak{P}}\) is a Frobenius element in the decomposition group \(D_{\mathfrak{P}} \subset \mathrm{Gal}(L/\mathbb{Q})\).
Remark 2 (cf. [1]). We remark on the following points to prevent possible misunderstandings regarding the notation of 1-([item:algebraic3]). Let \(p\) be a rational prime such that a prime \(\mathfrak{P}\mid p\) is unramified in \(L\). By the definition of \(g\colon\mathrm{Gal}(L/\mathbb{Q})\to L\), it follows that the residue class \(g(\mathrm{Frob}_{\mathfrak{P}})\bmod\mathfrak{P}\) is fixed by \(\mathrm{Frob}_{\mathfrak{P}}\). Therefore, we see that \[\begin{align} \nonumber g(\mathrm{Frob}_{\mathfrak{P}})\bmod\mathfrak{P}\in\mathbb{F}_{p}\subset\mathcal{O}_{L}/\mathfrak{P}. \end{align}\] In addition, we note that the residue class \(g(\mathrm{Frob}_{\mathfrak{P}})\bmod\mathfrak{P}\) is independent of the choice of \(\mathfrak{P}\mid p\) (see [2]).
The above theorem seems to differ from the classical definition of algebraic numbers. However, Rosen pointed out that all elements in \(\mathcal{P}_{\mathcal{A}}^{0}\) are the root of some non-zero polynomials:
Theorem 3 ([1]). If the element \(\boldsymbol{t}=(t_{p}\bmod p)_{p}\) is in \(\mathcal{P}_{\mathcal{A}}^{0}\), then, there exists a non-zero polynomial \(f(X)\in\mathbb{Q}[X]\) such that \(f(\boldsymbol{t})=0\) in \(\mathcal{A}\), and every such \(f(X)\) has a “rational root”.
In 3, we are inevitably led to focus on the part referred to as rational root. We may also regard the set \[\begin{align} \nonumber \mathcal{C}_{\mathcal{A}}=\{\boldsymbol{t}\in\mathcal{A}\;|\;{}^{\exists}f(X)\in\mathbb{Q}[X]\setminus\{0\}\;\text{s.t.}\;f(\boldsymbol{t})=0\;\text{in}\;\mathcal{A}\} \end{align}\] as being more appropriate for the definition of finite algebraic numbers. However, it is known that \(\mathcal{C}_{\mathcal{A}}\) does not possess the properties of \(\overline{\mathbb{Q}}\) in the following respects:
Proposition 4 ([1], [3]). The following holds.
\(\mathcal{C}_{\mathcal{A}}\) is uncountable set.
\(\mathbb{Q}\subsetneq\mathcal{P}_{\mathcal{A}}^{0}\subsetneq\mathcal{C}_{\mathcal{A}}\subsetneq\mathcal{A}\) holds.
Furthermore, it is also known that the elements of \(\mathcal{P}_{\mathcal{A}}^{0}\) are obtained via the \(\mathcal{A}\)-valued Frobenius automorphism \[\begin{align} \nonumber F_{\mathcal{A}}\colon H^{0}_{dR}(\mathrm{Spec}(L))\otimes\mathcal{A}\xrightarrow{\sim} H^{0}_{dR}(\mathrm{Spec}(L))\otimes\mathcal{A}, \end{align}\] where \(L\) runs over the finite Galois extensions over \(\mathbb{Q}\) (cf. [1]). Therefore, \(\mathcal{P}_{\mathcal{A}}^{0}\) admits some geometric considerations (the “\(0\)” of \(\mathcal{P}_{\mathcal{A}}^{0}\) comes from the “\(0\)”-th cohomology).
It is also an interesting problem to give an element of \(\mathcal{A}\) which is not contained in \(\mathcal{P}_{\mathcal{A}}^0\). We call those elements \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental numbers. The first attempt of the following theorem has been given by Luca–Zudilin ([4]):
Theorem 5 ([4]). Define the \(q\)-Fibonacci sequence \((F_{n}(q))_{n}\in\left(\mathbb{Z}[q]\right)^{\mathbb{N}}\) by \(F_{0}(q)=0,\;F_{1}(q)=1\) and \[\begin{align} \nonumber F_{n}(q)=F_{n-1}(q)+q^{n-2}F_{n-2}(q),\quad {}^{\forall}n\in\mathbb{Z}_{\geq2}. \end{align}\] Then, \((F_{p}(q)\bmod p)_{p}\) is \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental for all \(q\in\mathbb{Z}_{>1}\).
Remark 6 (cf. [3], [5]). Anzawa–Funakura’s result ([3]) is also important. They proved the “\(\mathcal{C}_{\mathcal{A}}\)-transcendence” of \((F_{p}(q)\bmod p)_{p}\) for all square-free \(q\), under generalized Riemann hypothesis, earlier than Luca–Zudilin ([4]). Here, we note that the two results should be understood separately.
For the research of \(\mathcal{C}_{\mathcal{A}}^{0}\)-transcendence in particular, the reader is referred to the paper of Matsusaka and Seki ([5]). The authors succeeded in proving that \((F_{p}(q)\bmod p)_{p}\) is \(\mathcal{C}_{\mathcal{A}}\)-transcendental for all \(q\in\mathbb{Z}_{>1}\). It is a generalization of both results in [3], [4]. Matsusaka and Seki also carried out various other investigations about \(\mathcal{C}_{\mathcal{A}}\)-transcendence in [5].
After 5, Luca and Zudilin obtained another \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental element in the case of \(\dim=1\) algebraic varieties as follows:
Theorem 7 ([6]). Let \(E\) be an elliptic curve over \(\mathbb{Q}\) without complex multiplication. Then the sequence of Frobenius traces \(\boldsymbol{a}=(a_{p}(E)\bmod p)_{p}\) is \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental, where \(a_p(E):=p+1-\# E(\mathbb{F}_p)\) for all \(p\) such that \(E\) has good reduction at \(p\).
Based on the above results, we could remove the condition “complex multiplications”. Here is the first main theorem.
Main Theorem 1. Let \(E\) be an elliptic curve over \(\mathbb{Q}\). Then, \(\boldsymbol{a}=(a_{p}(E)\bmod p)_{p}\) is \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental.
Comparing to the study in Luca–Zudilin ([6]), we can assure that 1 is a generalization of [6].
Remark 8. Although 1 overlaps with [5] and both cases require the Sato–Tate conjecture in order to include the CM case, both proofs are independent of each other. More precisely, [5] is based on the Luca–Zudilin criterion ([5]), whereas the present paper derives a contradiction by considering only rational primes that split completely in some finite Galois extensions over \(\mathbb{Q}\). We emphasize that the present paper is based on a different perspective from [5].
Furthermore, in this paper, we give an another \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental element as follows:
Main Theorem 2. Let \(A\) be a principally polarized abelian variety over \(\mathbb{Q}\). Suppose that the \(\ell\)-adic Galois representation1 \[\begin{align} \nonumber \rho_{A,\ell}\colon\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow \mathrm{GSp}_{2g}(\mathbb{Z}_{\ell}) \end{align}\] is surjective. 2 Then the sequence of Frobenius traces \(\boldsymbol{a}=(a_{p}\bmod p)_{p}=(\mathrm{tr}(\rho_{A,\ell}(\mathrm{Frob}_{p}) \bmod p)_{p}\) is \(\mathcal{P}_{\mathcal{A}}^{0}\)-transcendental.
This work originated from an informal private seminar held under the kind guidance of Professor Takuya Yamauchi at Tohoku University. The author would like to express his deepest gratitude to Professor Yamauchi for providing the opportunity to discuss this work and for his fruitful advice and insightful remarks.
The author is also deeply grateful to Professors Toshiki Matsusaka, Shin-ichiro Seki, Yasuo Ohno, and Wadim Zudilin for their careful reading of the manuscript and for many helpful suggestions. Finally, the author would like to thank Tohoku University for its incredible hospitality during the course of this research project.
In this section, we prove 1. In [6], Luca and Zudilin proved 7 by using Serre’s open image theorem stated as follows:
Theorem 9 ([9]). For any non-CM elliptic curve \(E\) over \(\mathbb{Q}\), the \(\ell\)-adic Galois representation \[\begin{align} \nonumber \rho_{E,\ell}\colon\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{Aut} _{\mathbb{Z}}(E[\ell])\simeq\mathrm{GL}_{2}(\mathbb{F}_{\ell}) \end{align}\] is surjective for all but finitely many primes \(\ell\).
Our proof proceeds without using the above theorem, focusing instead on primes that split completely in some finite Galois extensions over \(\mathbb{Q}\) and applying Sato–Tate conjecture. Here we summarize the result. The CM case is due to Hecke and we refer to [10]. The non-CM case is due to a celebrated work [11]. For an elliptic curve \(E/\mathbb{Q}\) and a prime \(p\) such that \(E\) has good reduction at \(p\), we choose \(\theta_p\in [0,\pi]\) satisfying \(\cos{\theta_{p}}:=a_{p}(E)/2\sqrt{p}\). We call such a prime \(p\) a good prime of \(E\).
Theorem 10 (Sato–Tate conjecture, [10], [11]). Let \(E\) be an elliptic curve over \(\mathbb{Q}\). Then, the following holds.
When \(E\) is non-CM, then for \(0\leq \alpha<\beta\leq\pi\), it holds \[\begin{align} \nonumber \lim_{X\to\infty}\frac{\#\left\{ p\leq X\;|\;\text{p is a good prime of E},\;\alpha\leq \theta_{p}\leq\beta\right\}}{\#\left\{ p\;|\;p\leq X \right\}}=\frac{2}{\pi}\int_{\alpha}^{\beta}\sin^{2}{\theta}d\theta. \end{align}\]
When \(E\) has CM, then for \(0\leq \alpha<\beta\leq\pi\), it holds \[\begin{align} \lim_{X\to\infty}\frac{\#\left\{ p\leq X\;|\;\text{p is a good prime of E},\;\alpha\leq \theta_{p}\leq\beta\right\}}{\#\left\{ p\;|\;p\leq X \right\}}=\frac{\delta_{0}}{2}+\frac{1}{2\pi}\int_{\alpha}^{\beta}d\theta. \end{align}\] Here, \(\delta_{0}=\begin{cases} 1,\;\mathrm{if}\;\pi/2\in[\alpha,\beta],\\ 0,\;\mathrm{otherwise}. \end{cases}\)
proof of 1. We only consider infinitely many good primes. We assume \(\boldsymbol{a}=(a_{p}(E)\bmod p)_{p}\in\mathcal{P}_{\mathcal{A}}^{0}\) (and consequently, deduce a contradiction). By using the formulation in 1-([item:algebraic3]), there exists a finite Galois extension \(L/\mathbb{Q}\) of degree \(d=[L:\mathbb{Q}]\), a map \(g\colon \mathrm{Gal}(L/\mathbb{Q})\to L\), and finite elements \(b_1,\ldots,b_k\in L\) such that \[\begin{align} \nonumber \text{Im}(g)=\{b_{1},\ldots,b_{k}\}\;\text{and}\;a_{p}(E)\equiv g(\mathrm{Frob}_{\mathfrak{P}})=b_{i}\bmod \mathfrak{P}, \end{align}\] for some \(i\) with \(1\le i\le k\). By discarding \(b_i\) if necessary, we may assume that \(L\) is the Galois closure of \(\mathbb{Q}(b_1,\ldots,b_k)\). We set \[\begin{align} \nonumber S_{1}=\left\{ p\;|\;p\;\text{is completely splited in}\;L \right\}. \end{align}\] Then the natural density is given as \(\text{den}(S_{1})=d^{-1}>0\) by Chebotarev density theorem. It is important to note that there are infinitely many primes that belong to \(S_{1}\).
We fix such \(p\) and \(\mathfrak{P}\mid p\). The condition \(p\in S_{1}\) says \(\mathrm{Frob}_{\mathfrak{P}}=\text{id}\in\mathrm{Gal}(L/\mathbb{Q})\). Moreover, we have \(b:=g(\text{id})\in\mathbb{Q}\) by the definition of \(g\). We can assume that \(b\neq0\). Indeed, if \(E\) is non-CM, then the density of \(p\) such that \(a_{p}=0\) is \(0\), also if \(E\) is CM, then the density of such prime is \(1/2\). In either case, there exist infinitely many primes \(p\) for which \(a_{p}\neq0\). Since \(a_{p}\in\mathbb{Z}\) and \(b\in\frac{1}{N}\mathbb{Z}\) \(({}^{\exists}N\in\mathbb{Z}_{\gg0})\), we have \[\begin{align} \label{cong32a95p61b} a_{p}\equiv b\bmod p. \end{align}\tag{1}\] Using 1 and Hasse bound (cf. [12]), we obtain \[\begin{align} \label{eneq32p} \frac{p}{N}\leq |a_{p}-b| \leq 2\sqrt{p}+|b|. \end{align}\tag{2}\] 2 says that \(p\) lives in \(p<K:=(N+\sqrt{N^{2}+N|b|})^{2}\). Note that \(K\) is independent of the choice of \(p\), so we have \[\begin{align} \label{a95p61b} a_{p}-b=0,\;{}^{\forall}p\in S_{1}\;\text{and}\;p>K. \end{align}\tag{3}\] We define another sets by \[\begin{align} \nonumber & S_{2}=\{ p\;|\;p\;\text{is completely splited in}\;L,\;b\in\mathbb{Z}_{p},\;p>K,\;a_{p}=b \},\\ \nonumber & S_{3}=\{ p\;|\;b\in\mathbb{Z}_{p},\;p>K,\;a_{p}=b \}. \end{align}\] Then, we have \(S_{2}\subseteq S_{3}\).
(i) \(E\) : non-CM elliptic curve
For any \(K^{\prime}>K\) and \(\varepsilon_{K^{\prime}}>0\) such that \(\varepsilon_{K^{\prime}}\to0\) as \(K^{\prime}\to\infty\), the equality \[\begin{align} \nonumber
\text{den}(S_{3})=\lim_{\varepsilon_{K^{\prime}}\to0}\frac{2}{\pi}\int_{\frac{\pi}{2}-\varepsilon_{K^{\prime}}}^{\frac{\pi}{2}+\varepsilon_{K^{\prime}}}\sin^{2}\theta \mathrm{d}\theta=0
\end{align}\] holds by Sato–Tate conjecture (10-([item:Sato--Tate32nonCM])).
(ii) \(E\) : CM elliptic curve
By the condition \(b\neq0\), we have the same equation \[\begin{align}
\nonumber \text{den}(S_{3})=0
\end{align}\] from 10-([item:Sato--Tate32CM]).
However, in both cases, it contradicts \[\begin{align} \nonumber S_{2}\subseteq S_{3}\;\text{and}\;\text{den}(S_{2})=\text{den}(S_{1})>0. \end{align}\] Thus, we obtain the conclusion. ◻
In this section, we prove 2 using the proof method in [6]. Note that the condition in 2 are imposed in order to apply the next theorem by Serre ([13]) as follows:
Theorem 11 ([13]). Let \(A\) be a principally polarized abelian variety over a number field \(K\), such that
\(\mathrm{End}_{K}(A)=\mathbb{Z}\),
\(\dim_{K}(A)=g\) satisfies \(g\equiv 1\pmod2\) or \(g=2\) or \(g=6\).
Then, the \(\ell\)-adic Galois representation \(\rho_{A,\ell}\) is surjective for all but finitely many primes \(\ell\).
proof of 2. Suppose that \(\boldsymbol{a}=(a_{p}\bmod p)_{p}\in\mathcal{P}_{\mathcal{A}}^{0}\). Let us choose a prime \(\ell\) such that the Galois representation \(\rho_{A,\ell}\) is surjective. Since \(\mathrm{GSp}_{2g}\) is smooth over \(\mathbb{Z}_\ell\), for each \(m\ge 1\), the mod \(\ell^m\) map \(\mathrm{GSp}_{2g}(\mathbb{Z}_\ell)\to \mathrm{GSp}_{2g}(\mathbb{Z}_l/\ell^m \mathbb{Z}_\ell)\) is surjective. Thus, the \(\bmod \ell^{m}\)-representation \[\begin{align} \nonumber \overline{\rho}_{A,\ell^m}\colon\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to\mathrm{GSp}_{2g}(\mathbb{Z}/\ell^{m}\mathbb{Z}) \end{align}\] is also surjective for any \(m\geq 1\).
Let \(K_{m}\) be the Galois extension which corresponds to \(\ker\overline{\rho}_{A,\ell^m}\). Then we have \[\begin{align} \nonumber \mathrm{GSp}_{2g}(\mathbb{Z}/\ell^{m}\mathbb{Z})\simeq \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\Big/\ker\overline{\rho}_{\ell,A}\simeq \mathrm{Gal}(K_{m}/\mathbb{Q}). \end{align}\] Note that the first isomorphism holds by surjectivity of \(\rho_{A,\ell}\).
Now we consider the following sets \[\begin{align} \nonumber & S=\left\{ p\;|\;a_{p}=0 \right\},\\ \nonumber & S_{m}=\left\{ p\;|\;a_{p}\equiv 0\pmod {\ell^{m}} \right\}, \end{align}\] then \(S\subseteq S_{m}\) holds clearly. Comparing the density, we have \[\begin{align} \nonumber \text{den}(S)\leq \text{den}(S_{m})=\frac{\#\{ \sigma\in\mathrm{Gal}(K_{m}/\mathbb{Q})\;|\;\mathrm{tr}(\overline{\rho}_{\ell,A}(\sigma))=0 \}}{\# \mathrm{Gal}(K_{m}/\mathbb{Q})}\leq\frac{\#H_{m}}{\#\mathrm{GSp}_{2g}(\mathbb{Z}/\ell^{m}\mathbb{Z})}, \end{align}\] where, \(H_{m}=\{ M\in\mathrm{GSp}_{2g}(\mathbb{Z}/\ell^{m}\mathbb{Z})\;|\;\mathrm{tr}M=0 \}\). For any sufficiently large \(m\ge 1\), we have the following estimate \[\begin{align} \nonumber \#\mathrm{Gal}(K_{m}/\mathbb{Q})=\#\mathrm{GSp}_{2g}(\mathbb{Z}/\ell^{m}\mathbb{Z})\approx \ell^{md},\quad d:=\dim \mathrm{GSp}_{2g}. \end{align}\] In addition, the dimension of hypersurface of \(\mathrm{GSp}_{2g}\) defined by \(\mathrm{tr}=0\) is \(d-1\). Then it holds \[\begin{align} \nonumber \# H_{m}\ll\ell^{m(d-1)} \end{align}\] by [14]. Therefore, we obtain \[\begin{align} \nonumber \text{den}(S_{m})\ll\frac{1}{\ell^{m}},\quad {}^{\forall}m\geq1, \end{align}\] and then, \(\text{den}(S)=0\) holds.
Hereafter, we always consider the rational prime \(p\) such that \(a_{p}\neq0\). For sufficiently large \(X\), define the set \[\begin{align} \nonumber T=\left\{ p\leq X\;|\;a_{p}\neq0,\;\ell\mid a_{p} \right\}. \end{align}\] The latter condition means that the Frobenius is sent to zero-trace matrices in \(\text{Im}\overline{\rho}_{\ell,A}=\mathrm{GL}_{2g}(\mathbb{F}_{\ell})\). Then \(\text{den}(T)>0\) holds by Chebotarev density theorem, and obtain \[\begin{align} \label{eneq:32prime1} \#T\gg\frac{X}{\log X}. \end{align}\tag{4}\] Let \(L/\mathbb{Q}\) and \(\{ b_{1},\ldots,b_{k} \}\subset L\) denote the notation used in the proof in 1. In this situation, \(p\mid N_{L/\mathbb{Q}}(a_{p}-b_{i})\) and \(a_{p}-b_{i}\neq0\) hold. By \(p\leq X\) and Hasse–Weil bound \(|a_{p}|\leq 2g\sqrt{p}\), the following inequality \[\begin{align} \nonumber |N_{L/\mathbb{Q}}(a_{p}-b_{i})|\leq (2g\sqrt{X}+|b_{i}|)^{[L:\mathbb{Q}]} \end{align}\] implies that \(N_{L/\mathbb{Q}}(a_{p}-b_{i})\) has size \(X^{O(1)}\), so that the number of prime factors of \(N_{L/\mathbb{Q}}(a_{p}-b_{i})\) is at most \(O(\log X)\). Since \(a_{p}\) has the order \(O(\sqrt{X})\) of choices, the number of rational primes coming from the divisibility of \(N_{L/\mathbb{Q}}(a_{p}-b_{i})\) is at most \[\begin{align} \label{eneq:32prime2} O(\sqrt{X}\log X). \end{align}\tag{5}\] However, 4 and 5 lead to a contradiction. ◻
In general, \(\rho_{A,\ell}\) is unramified for all but finitely many \(p\) by Serre and Tate ([7]). It is known that the characteristic polynomial of Frobenius has integral coefficients and does not depend on the choice of \(p\) (cf. [8]), so that \(\boldsymbol{a}=(\mathrm{tr}(\rho_{A,\ell}(\mathrm{Frob}_{p}))\bmod p)_{p}\in\mathcal{A}\) is well-defined.↩︎