Infinitely many primes with a fixed Frobenius field for an elliptic curve over $\mathbb{Q}$

Abstract: In 1987, Elkies proved the striking result that every elliptic curve over $\mathbb{Q}$ has infinitely many supersingular primes. Motivated by this theorem and its connection with the Lang-Trotter conjecture, we study the analogous problem for Frobenius fields of elliptic curves. For certain families of non-CM elliptic curves $E/\mathbb{Q}$ and imaginary quadratic fields $K$, we prove that there exist infinitely many primes $p$ for which the Frobenius field of $E$ at $p$ equals $K$. More precisely, letting $π<em>E(x,K)$ denote the number of such primes with $p\leq x$, we establish the unconditional bound $π_E(x, K)\gg</em>{E, K, ε} (\log\log x)<sup>{1-ε}$ for every $ε&gt;0$. We also prove unconditional power-saving upper bounds for a restricted counting function associated with $π_E(x,K)$. The approach combines Deuring's theory of complex multiplication, properties of singular moduli, and arithmetic intersection theory on modular curves.

• The paper proves infinitely many primes with a prescribed imaginary quadratic Frobenius field for an explicit positive-density family of elliptic curves and squarefree integers, with the lower bound π_E(x) ≫ (log log x)^{1−ε}.
• The paper uses quadratic twists, Deuring’s lifting theorem, Hilbert class polynomials, singular-moduli estimates, and a Euclid-style iteration to construct distinct primes with the desired Frobenius field.
• The paper establishes a restricted upper bound of ≪_E √Δ(log Δ)x^{2δ} log log x and derives an unconditional lower bound for endomorphism-ring discriminants, while leaving the conjectured √x/log x growth unresolved.

Context and motivation

For a non-CM elliptic curve $E/\mathbb{Q}$ of conductor $N_E$ and a prime $p\nmid N_E$ of good reduction, the Frobenius trace $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$ determines the Frobenius field $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$, an imaginary quadratic field. Lang and Trotter conjectured in 1976 that for fixed squarefree $\Delta\geq 1$, the counting function

$$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$

satisfies $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$. While unconditional upper bounds have been progressively refined — from Serre's $x/(\log x)^\gamma$, through Cojocaru–Fouvry–Murty's $x(\log\log x)^{13/12}/(\log x)^{25/24}$ and Zywina's smoothed Chebotarev bound, to Thorner–Zaman's $N_E$0 — no unconditional lower bound of any kind was previously known for this function, even for a single pair $N_E$1. The only comparable result is Elkies' 1987 theorem on the infinitude of supersingular primes (the case $N_E$2 of the analogous trace problem), quantified by Fouvry–Murty as $N_E$3.

The paper under review, by Tian Wang (2608.17639), establishes the first nontrivial unconditional lower bounds for $N_E$4, valid for a positive-density family of pairs $N_E$5, together with a new power-saving upper bound for a restricted variant of the counting function.

Main results

The two principal theorems are as follows. Lower bound: let $N_E$6 be an elliptic curve whose $N_E$7-invariant $N_E$8 is an even integer, such that every odd prime $N_E$9 has Kodaira symbol $p\nmid N_E$0; let $p\nmid N_E$1 be squarefree, composite, with $p\nmid N_E$2. Then there are infinitely many primes $p\nmid N_E$3 with $p\nmid N_E$4, and more precisely

$p\nmid N_E$5

The hypotheses are not vacuous: a positive-density set of squarefree $p\nmid N_E$6 satisfies them, and any quadratic twist of the curve 128.a1 by odd squarefree integers provides admissible $p\nmid N_E$7. Upper bound: for non-CM $p\nmid N_E$8, squarefree $p\nmid N_E$9, and any $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$0, defining

$a_p(E)=p+1-\#E_p(\mathbb{F}_p)$1

where $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$2 is the conductor of the reduction's endomorphism ring, one has

$a_p(E)=p+1-\#E_p(\mathbb{F}_p)$3

At $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$4 the restricted function coincides with $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$5 itself; the theorem thus falls short of a power-saving bound for the full counting function, a point the author states plainly. The corollary deduced from it is an unconditional analogue of a GRH-conditional result of Cojocaru–Fitzpatrick on endomorphism ring discriminants: for almost all primes,

$a_p(E)=p+1-\#E_p(\mathbb{F}_p)$6

improving Schoof's unconditional $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$7 to a bound proportional to the Hasse quantity $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$8.

The lower bound: Euclid-type iteration with singular moduli

The proof adapts Elkies' Euclidean strategy to ordinary primes. Three structural reductions organize the argument. First, quadratic twists preserve Frobenius fields at all good primes outside the conductors, so one may replace $a_p(E)=p+1-\#E_p(\mathbb{F}_p)$9 by a convenient twist. Second, since supersingular primes contribute $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$0 (or $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$1, $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$2, $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$3 at $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$4), the assumption that $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$5 is composite forces all but at most one contributing prime to be ordinary. Third, the Kodaira hypothesis $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$6 at each odd bad prime guarantees, via Tate's algorithm and a local quadratic character, a twist of $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$7 with good reduction there.

The engine is Deuring's lifting theorem combined with Hilbert class polynomials $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$8: if $$\mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{a_p(E)^2-4p})$$9 is ordinary and $\Delta\geq 1$0 with $\Delta\geq 1$1, then $\Delta\geq 1$2, hence $\Delta\geq 1$3. Given a finite set $\Delta\geq 1$4 of known primes in $\Delta\geq 1$5, the author shows that every sufficiently large prime $\Delta\geq 1$6 is admissible: $\Delta\geq 1$7 and none of the primes in $\Delta\geq 1$8 divides it (otherwise Lemma on Deuring lifting would force $\Delta\geq 1$9, contradicting coprimality). A prime divisor $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$0 of $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$1 then yields a new prime of $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$2 after passing to a suitable quadratic twist; distinct admissible $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$3 yield distinct new primes. Iteration proves infinitude, contradicting a uniform bound $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$4 valid across all twists.

Two analytic inputs deserve emphasis. The key new lemma controls sums of $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$5 over singular moduli $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$6 of discriminant $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$7 lying within unit distance of $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$8: the sum is bounded below by $$\pi_E(x, \mathbb{Q}(\sqrt{-\Delta})):=\#\{p\le x:\ p\nmid N_E,\ \mathbb{Q}(\pi_p(E_p))=\mathbb{Q}(\sqrt{-\Delta})\}$$9 for large $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$0. Its proof combines a trivial height lower bound for differences of singular moduli, a counting estimate for CM points near $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$1 (via Autissier-type equidistribution and the bound $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$2), and Siegel's ineffective class number lower bound — which renders the constant $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$3 ineffective, though effective under GRH for Dirichlet $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$4-functions. From this, the paper derives that $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$5 for all sufficiently large $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$6, an effective special case of a finiteness theorem of Aslanyan et al. whose general proof is ineffective. Combining the split of the product into factors near and far from $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$7, one obtains $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$8 asymptotically.

Quantitatively, choosing all admissible primes $$\pi_E(x,\mathbb{Q}(\sqrt{-\Delta}))\sim C(\Delta,E)\sqrt{x}/\log x$$9 in $x/(\log x)^\gamma$0 produces $x/(\log x)^\gamma$1 distinct new primes, and the height bound $x/(\log x)^\gamma$2 gives a recursive growth relation $x/(\log x)^\gamma$3. Iterating yields $x/(\log x)^\gamma$4 primes up to $x/(\log x)^\gamma$5. Note the exponent here is stronger than Fouvry–Murty's triple-logarithmic bound for supersingular primes, though for a restricted family of $x/(\log x)^\gamma$6 rather than all curves.

The upper bound: arithmetic intersection theory on $x/(\log x)^\gamma$7

The upper-bound argument departs from the Chebotarev-plus-sieve paradigm of prior work. Every prime counted by $x/(\log x)^\gamma$8 divides $x/(\log x)^\gamma$9 for some $x(\log\log x)^{13/12}/(\log x)^{25/24}$0, so bounding reduces to controlling prime divisors of these values. The author considers the horizontal divisors $x(\log\log x)^{13/12}/(\log x)^{25/24}$1 (Zariski closure of $x(\log\log x)^{13/12}/(\log x)^{25/24}$2) and $x(\log\log x)^{13/12}/(\log x)^{25/24}$3 (the CM divisor $x(\log\log x)^{13/12}/(\log x)^{25/24}$4) on the coarse moduli scheme $x(\log\log x)^{13/12}/(\log x)^{25/24}$5, and applies Charles' asymptotic formula for global intersection numbers:

$x(\log\log x)^{13/12}/(\log x)^{25/24}$6

The Hecke orbit $x(\log\log x)^{13/12}/(\log x)^{25/24}$7 decomposes via CM theory into CM divisors $x(\log\log x)^{13/12}/(\log x)^{25/24}$8 weighted by representation numbers $x(\log\log x)^{13/12}/(\log x)^{25/24}$9 counting cyclic ideals. At each counted prime $N_E$00, the local intersection multiplicity satisfies $N_E$01 by positivity, while the archimedean contribution is bounded using the $N_E$02-expansions of $N_E$03 and $N_E$04, giving $N_E$05. Summing Charles' formula over $N_E$06 and comparing the resulting lower and upper bounds yields $N_E$07; taking $N_E$08 and adding the $N_E$09 small-prime contribution gives Theorem 2. The author notes the improvement over the "trivial" bound $N_E$10 is modest, and that when $N_E$11 grows polynomially in $N_E$12 the trivial bound can even be slightly better.

Limitations and open questions

Several restrictions are acknowledged explicitly. The lower bound requires the congruence $N_E$13 (ensuring $N_E$14 splits in $N_E$15, which keeps $N_E$16), compositeness of $N_E$17, integrality and parity of $N_E$18, and Kodaira type $N_E$19 at odd bad primes. Removing these conditions would require showing that $N_E$20 is not an $N_E$21-unit for singular moduli $N_E$22 and finite sets $N_E$23 containing the prime divisors of $N_E$24 — a statement the author reports as unavailable in the current literature. The implicit constant in the lower bound is ineffective due to Siegel's theorem, becoming effective under GRH. On the upper-bound side, achieving a power saving for the unrestricted $N_E$25 would require either $N_E$26 (not expected to hold generally) or a proof that almost all integers in the relevant conductor set have uniformly bounded numbers of prime factors; the author observes that optimal bounds in the non-archimedean intersection inequality or in $N_E$27 would potentially give $N_E$28, matching the Lang–Trotter order of magnitude up to $N_E$29.

Conclusion

This paper supplies the first unconditional lower bounds for primes with prescribed Frobenius field, proving $N_E$30 for a family of non-CM elliptic curves and imaginary quadratic fields satisfying explicit arithmetic conditions, via a Euclid-style iteration built on Hilbert class polynomials, quadratic twisting, and new estimates for values of class polynomials at integral $N_E$31-invariants. Complementarily, it introduces arithmetic intersection theory on $N_E$32 as a tool for upper bounds on restricted versions of the counting function, yielding power savings in $N_E$33 and an unconditional almost-all lower bound for endomorphism ring discriminants of the form $N_E$34. The gap between these bounds and the conjectured $N_E$35 remains substantial, and its closure appears tied to open problems concerning $N_E$36-unit properties of differences between singular moduli and fixed algebraic numbers.