Construction Of Non-Odd Galois Representations With Large Image

In this work, we construct Galois representations with large image that are not GL r -odd (or, equivalently, not regular at infinity). More precisely, for every prime p {\v e} 3, every integer r {\v e} 2, and every integer a P rp1 'p'1q r q{2, r '2s satisfying a '' r pmod 2q, we construct a continuous Galois representation $\rho$\,: G Q __ GL r pQ p q whose image is commensurable with GL r pZ p q and satisfies |trp$\rho$pcqq| '' a, where c denotes a complex conjugation. We also obtain analogous results for the orthogonal group O r pQ p q.