The electronic structure of C_(4N+2) carbon rings exhibits competing many-body effects of Huckel aromaticity, second-order Jahn-Teller and Peierls instability at large sizes. This leads to possible ground state structures with aromatic, bond angle or bond length alternated geometry. Highly accurate quantum Monte Carlo results indicate the existence of a crossover between C_10 and C_14 from bond angle to bond length alternation. The aromatic isomer is always a transition state. The driving mechanism is the second-order Jahn-Teller effect which keeps the gap open at all sizes.