Joint Sato–Tate Localization near the Cramér Scale in Growing Arithmetic Progressions

Abstract

Let $E_1, E_2 / \mathbf{Q}$ be fixed twist-inequivalent non-CM elliptic curves. This paper studies the joint Sato–Tate distribution of the Frobenius angles $$(\theta_{1,p}, \theta_{2,p}) \in [0,\pi]^2$$ for primes constrained simultaneously by a literal moving interval $$X < p \le X+h$$ and a reduced residue class $$p \equiv a \pmod q,$$ where the modulus $q=q(X)$ is allowed to grow. The analytic input is a finite bidegree family of Dirichlet-twisted symmetric-power Rankin–Selberg $L$-functions $$L\!\left( s, \operatorname{Sym}^m \pi_1 \times (\operatorname{Sym}^n \pi_2 \otimes \chi^\ast) \right), \qquad 0 \le m,n \le M,$$ together with the scalar Dirichlet packet required for the denominator. Under GRH for the finite packet used at the chosen spectral depth, a two-scale plateau explicit formula, followed by Dirichlet recombination before prime-power and hard-edge removal, gives $$D^{(2)}_{X,h;q,a} \ll_{E_1,E_2} \frac{1}{M} + M^2 \mathcal{R}^{(2),\mathrm{AP}}_{M,q}(X,h,\Delta).$$ At the balanced edge $$\Delta = \varphi(q)\sqrt{X}\log X,$$ the natural phase variable is $$\mathfrak{G} = \frac{h}{\varphi(q)\sqrt{X}\log X}.$$ Writing $$\eta = \frac{1+\log \mathfrak{G}}{\mathfrak{G}} + \frac{h}{X\log X},$$ and choosing $$M \asymp \eta^{-1/3},$$ one obtains the supercritical joint discrepancy estimate $$D^{(2)}_{X,h;q,a} \ll_{E_1,E_2} \eta^{1/3}.$$ In particular, if $$h = \sqrt{X}(\log X)^A, \qquad A>1,$$ and $$\varphi(q) = o\!\left((\log X)^{A-1}\right),$$ then $$D^{(2)}_{X,h;q,a} \ll_{E_1,E_2,A} \left[ \frac{\varphi(q)}{(\log X)^{A-1}} \left( 1+ \log \frac{(\log X)^{A-1}}{\varphi(q)} \right) \right]^{1/3}.$$ Version v1.1 adds a critical Cramér-order occupancy theorem. For every fixed rectangle $$B = I_1 \times I_2 \subset [0,\pi]^2$$ of positive product Sato–Tate measure, there exist fixed constants $M_B$ and $G_B$ such that, under GRH for the corresponding fixed bidegree range and the required Dirichlet twists, every sufficiently large admissible window $$h = G\,\varphi(q)\sqrt{X}\log X, \qquad G \ge G_B,$$ contains $$N_B(X,h;q,a) \gg_{E_1,E_2,B} \frac{h}{\varphi(q)\log X}$$ primes satisfying both the arithmetic-progression condition and the prescribed joint Sato–Tate condition. At the threshold order $G=G_B$, this gives $$N_B(X,h;q,a) \gg_{E_1,E_2,B} \sqrt{X}.$$ Thus the paper exhibits a two-phase structure: $$\text{fixed spectral depth} \longrightarrow \text{critical Cramér-order positive occupancy},$$ while $$\mathfrak{G} \to \infty \quad \text{with growing spectral depth} \longrightarrow \text{full joint equidistribution}.$$ The $M^2$ harmonic truncation cost and the resulting $1/3$ exponent are not claimed as new joint Sato–Tate phenomena. The contribution is the compatibility of the established joint Rankin–Selberg/Sato–Tate architecture with literal Cramér-order and near-Cramér hard intervals, an individually growing arithmetic-progression modulus, and growing angular resolution. Version v1.1 strengthens the previous version by adding the critical Cramér-order occupancy theorem, the associated fixed-spectral-depth interpretation, and revised literature positioning. The underlying supercritical discrepancy theorem and Rankin–Selberg packet analysis are unchanged.