Since the AD 775 and AD 994 Δ14C peak (henceforth M12) was first measured by Miyake et al. (2012, 2013), several possible production mechanisms for these spike have been suggested, but the work of Mekhaldi et al. (2015) shows that a very soft energy spectrum was involved, implying that a strong solar energetic particle (SEP) event (or series of events) was responsible. Here we present Δ14C values from AD 721–820 Sequoiadendron giganteum annual tree-ring samples from Sequoia National Park in California, USA, together with Δ14C in German oak from 650–670 BC. The AD 721–820 measurements confirm that a sharp Δ14C peak exists at AD 775, with a peak height of approximately 15‰ and show that this spike was preceded by several decades of rapidly decreasing Δ14C. A sharp peak is also present at 660 BC, with a peak height of about 10‰, and published data (Reimer et al. 2013) indicate that it too was preceded by a multi-decadal Δ14C decrease, suggesting that solar activity was very strong just prior to both Δ14C peaks and may be causally related. During periods of strong solar activity there is increased probability for coronal mass ejection (CME) events that can subject the Earth’s atmosphere to high fluencies of solar energetic particles (SEPs). Periods of high solar activity (such as one in October–November 2003) can also often include many large, fast CMEs increasing the probability of geomagnetic storms. In this paper we suggest that the combination of large SEP events and elevated geomagnetic activity can lead to enhanced production of 14C and other cosmogenic isotopes by increasing the area of the atmosphere that is irradiated by high solar energetic particles.