Maize "was first domesticated from the wild teosinte grass in southern Mexico." Two companion 2009 papers in PNAS establish using radiocarbon dates that domesticated maize and squash starch were originally domesticated around 7000 BCE in Mesoamerica.
Specifically, by "around 7000 BC they see local hunter-gatherer groups in West Mexico as having effectively domesticated teosinte (creating maize), squash, and beans, thus creating the milpa system of agriculture that would become one of the hallmarks of Mesoamerican culture from then on." The individual plant domestications happen in different places, but really gain traction when they are assembled into a complete agricultural package that has sufficient nutrition to support a population of farmers.
Another Mesoamerican domesticate, chocolate, arrived in the Southwest much later, in the 8th century CE.
There had been a long standing debate prior to this study over whether maize then arrived in the American Southwest via the Mexican highlands or a coastal route. Now that all the facts are in, the answer is both. First, a highland route ca. 2100 BCE (around the time of Mexico's pre-Olmec civilization which was the earliest civilization of Mesoamerica and the earliest pre-Classical Mayan civilization), and then two thousand years later, ca. 0 CE, about 250 years before the beginning of Classical Mayan civilization. Southwestern maize was enriched by new strains that arrive via a lowland coastal route.
"When considered together, the results suggest that the maize of the U.S. Southwest had a complex origin, first entering the U.S. via a highland route about 4,100 years ago and later via a lowland coastal route about 2,000 years ago," said [co-author] Jeffrey Ross-Ibarra, an associate professor in the Department of Plant Sciences.The Southwestern variety that emerged had mutations adapting its received variety to drought and making it sweeter.
The new study used ancient DNA from corn cobs as much as 5,910 years old found at the archaeological sites (which also each provide an archaeologically calibration date limiting the need to over rely on mutation rate estimates) and DNA from traditional maize varieties in Mexico and the American Southwest to understand how maize.
The richness of the ancient and modern DNA samples relied upon was truly phenomenal for this kind of study, particularly for the sometimes less intensively studied New World crops (some references omitted without editorial indication):
Twenty-five archaeological maize cob samples from the Southwest United States dating from 4,300 to 740 years BP, three from Mexico dating from 5,910 to 1,410 BP, and four ancient Arica samples were obtained from the repositories and individuals listed in Supplementary Table 7 . . . In addition, previously published sequence data12 corresponding to an ancient sample from Mexico, was also used.References
With the exception of the Turkey House Ruin sample, all of the archaeological cob samples from the Southwest United States and Mexico were recovered from dry cave contexts, and the Chilean (Arica) samples came from the dry desert coast of South America. All of the archaeological samples were desiccated, uncarbonized and in an excellent state of preservation. The cobs recovered from sites in the Southwest United States fall into two distinct morphological and temporal categories. These two temporally separated and morphologically distinct forms of maize correlate quite closely with the structural analysis groupings based on aDNA. The early southwestern maize, including samples from McEuen and Bat Caves, and from the early occupation at Tularosa Cave (1,850–1,750 BP), variously labelled as ‘Chapalote’ or ‘small cob maize’4 is a small cob, small kernel form having a thick midsection (1.9–2.5 cm diameter) and tapered ends (Pineapple shape) and 10–12 rows of kernels. The maize from the later occupation at Tularosa Cave (700–900 BP), as well as the Turkey House Ruin sample (670 BP), is a larger cob, larger kernel form, having parallel sides (cylinder shape), eight to ten rows of kernels, and a much smaller diameter than the earlier form (1.3–1.6 cm).
Data for modern samples (maize landraces, Z. m. parviglumis and tripsacum) were obtained from the HapMap2 set and downloaded from Panzea's website (www.panzea.org). Additionally, we generated shotgun data from an individual from the highlands of northern Mexico.
 Rute R. da Fonseca, Bruce D. Smith, Nathan Wales, Enrico Cappellini, Pontus Skoglund, Matteo Fumagalli, José Alfredo Samaniego, Christian Carøe, María C. Ávila-Arcos, David E. Hufnagel, Thorfinn Sand Korneliussen, Filipe Garrett Vieira, Mattias Jakobsson, Bernardo Arriaza, Eske Willerslev, Rasmus Nielsen, Matthew B. Hufford, Anders Albrechtsen, Jeffrey Ross-Ibarra, M. Thomas P. Gilbert. "The origin and evolution of maize in the Southwestern United States." 1 Nature Plants 14003 (January 8, 2015) (some section of the paper are currently open access).
 Piperno, D., Ranere, A., Holst, I., Iriarte, J., & Dickau, R. "Starch grain and phytolith evidence for early ninth millennium B.P. maize from the Central Balsas River Valley", 106 (13) Mexico Proceedings of the National Academy of Sciences 5019-5024 (2009).
 Ranere, A., Piperno, D., Holst, I., Dickau, R., & Iriarte, J., "The cultural and chronological context of early Holocene maize and squash domestication in the Central Balsas River Valley", 106(13) Mexico Proceedings of the National Academy of Sciences 5014-5018 (2009).
 Zizumbo-Villarreal, D., & Colunga-GarcíaMarín, P., "Origin of agriculture and plant domestication in West Mesoamerica", Genetic Resources and Crop Evolution (2010).
 Washburn DK, Washburn WN, & Shipkova PA, "Cacao consumption during the 8th century at Alkali Ridge", 40 Southeastern Utah Journal of Archaeological Science 2007-2013 (2013).