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Construction Of Non-Odd Galois Representations With Large Image

Donghyeok Lim, Christian Maire

math.NTarXiv:2608.04577

Abstract

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 e 3, every integer r e 2, and every integer a P rp1 'p'1q r q2, r '2s satisfying a '' r pmod 2q, we construct a continuous Galois representation ρ\,: G Q _ GL r pQ p q whose image is commensurable with GL r pZ p q and satisfies |trpρpcqq| '' a, where c denotes a complex conjugation. We also obtain analogous results for the orthogonal group O r pQ p q.

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