Local Euler Factor Matching via Projected Orbital Integrals: From Selberg Kernels to L-Functions of Elliptic Curves

We prove the Local Euler Matching Proposition for non-CM modular elliptic curves on compact Shimura curves, resolving the central conditional gap in the projected trace formula program. Let E/ℚ be a non-CM modular elliptic curve of conductor N, with associated automorphic representation π_E on GL₂(𝔸). Let B/ℚ be a definite quaternion algebra of discriminant Δ_B coprime to N, and let π_B = JL(π_E) be the Jacquet–Langlands transfer.

For every finite place v, we construct an explicit admissible Selberg kernel f(π_B,v,s) such that the stable orbital integral satisfies

SO(γ_v, f(π_B,v,s)) = −(d/ds) log L_v(E,s)

at all unramified spherical places v ∤ NΔ_B, with explicit correction terms at the finitely many ramified places. We simultaneously prove the admissibility lemmas showing that the composed kernel

f(π_B,s) = f(π_B) ∗ Φ_s(Δ)

belongs to the Harish–Chandra Schwartz class for the compact Selberg trace formula. Together with the global arithmetic surface torsion results of the companion papers, this completes the analytic foundation of the non-CM Euler product identity

log T_A^(2)(π_B) = −[(d/ds) log L(E,s)] at s = 1 + κ_E,

where κ_E is an explicit archimedean–ramification correction.