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 proves a conditional joint Sato–Tate theorem for primes constrained simultaneously to a literal moving interval $(X, X+h]$ and a reduced residue class $a \pmod q$, with the modulus $q = q(X)$ allowed to grow. The analytic input is a finite bidegree family of Dirichlet-twisted symmetric-power Rankin–Selberg $L$-functions $$L\left( s, \operatorname{Sym}^m \pi_1 \times (\operatorname{Sym}^n \pi_2 \otimes \chi^*) \right), \qquad 0 \le m, n \le M,$$ under GRH. A two-scale plateau explicit formula, followed by Dirichlet-character recombination before hard-edge removal, yields a finite-depth joint rectangle discrepancy estimate of the form $$D^{(2)}_{X,h;q,a} \ll_{E_1,E_2} \frac{1}{M} + M^2 \mathcal R^{(2),\mathrm{AP}}_{M,q}(X,h,\Delta).$$ At the balanced edge $$\Delta \asymp \varphi(q)\sqrt X\log X,$$ the natural Cramér-normalized 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 the angular truncation depth at the scale $M \asymp \eta^{-1/3}$, one obtains $$D^{(2)}_{X,h;q,a} \ll_{E_1,E_2} \eta^{1/3}.$$ In particular, for $$h = \sqrt X(\log X)^A, \qquad A > 1,$$ and $$\varphi(q) = o\left((\log X)^{A-1}\right),$$ the discrepancy satisfies $$D^{(2)}_{X,h;q,a} \ll_{E_1,E_2,A} \left[ \frac{\varphi(q)}{(\log X)^{A-1}} \left( 1 + \log \frac{(\log X)^{A-1}}{\varphi(q)} \right) \right]^{1/3}.$$ The result combines near-Cramér hard-window localization, an individually growing arithmetic-progression modulus, and genuinely two-dimensional joint angular localization. The Rankin–Selberg conductor and zero-count bookkeeping remains linear in the bidegree $d_{m,n} = (m+1)(n+1)$, so no additional bidegree loss occurs beyond the standard two-dimensional $M^2$ truncation cost. This work is a distinct successor to two preceding hard-window Sato–Tate papers by the author. Quantitative Sato–Tate in Near-Square-Root Prime Intervals under Symmetric-Power GRH (Zenodo v3.0, DOI 10.5281/zenodo.22112048) provides the single-curve near-square-root finite-depth localization prototype. Near-Cramér Sato–Tate Equidistribution in Growing Arithmetic Progressions (Zenodo v1.0r1, DOI 10.5281/zenodo.22138400) introduces the growing-modulus arithmetic-progression realization and the recombination-before-hard-edge mechanism. The present paper carries this hard-window framework into the joint Sato–Tate setting through a growing bidegree Rankin–Selberg packet. It is a separate research article and is not a new version of either preceding paper.