Abstract: Abstract Personalizing large language models (LLMs) is essential for delivering AI assistance that aligns with individual users’ styles, intents, and preferences. While per-user fine-tuning can substantially enhance personalization quality, it introduces significant parameter and storage overhead, limiting scalability to large user populations. We propose \textbf{PLUME} (Personalized Low-Rank Adaptation through Modular and Shared Subspace), a lightweight framework that achieves efficient and expressive per-user adaptation by leveraging a shared task-specific subspace. PLUME enables scalable personalization by maintaining user-specific expressiveness while reducing per-user parameters by an order of magnitude. Specifically, PLUME first learns a global task subspace from aggregated user data. Personalization is then achieved by training only a lightweight small square matrix within this subspace, enabling each user to obtain a tailored model while keeping shared components fixed. Cross-layer shared parameters and rank-1 residual terms are further introduced to significantly reduce redundancy while maintaining expressiveness. Experiments on multiple personalized text generation benchmarks demonstrate that PLUME achieves comparable or superior performance to strong baselines, while reducing per-user parameters by over 95\%. These results establish shared-subspace modulation with minimal residuals as a scalable and semantically grounded approach to LLM personalization. 