Browsing by Author "Alvarado Torres, Matías Nicolás"

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    Arithmetic of Drinfeld modules
    (2025) Alvarado Torres, Matías Nicolás; Pasten Vásquez, Héctor Hardy; Pontificia Universidad Católica de Chile. Facultad de Matemáticas
    Drinfeld modules, introduced by Vladimir Drinfeld in the 1970s, have become acornerstone in the arithmetic of global function fields. These objects serve as thefunction field analogues of elliptic curves and abelian varieties, but with a structure thatis uniquely adapted to the arithmetic of positive characteristic. Defined over rings offunctions rather than number fields, Drinfeld modules allow for the development of arich arithmetic theory that mirrors, and in many ways extends, the classical theory ofelliptic curves. Their moduli spaces, Galois representations, and associated L-functionshave all been studied extensively, revealing deep analogies with the number field caseand offering new phenomena unique to the function field setting. From an arithmeticstandpoint, Drinfeld modules provide explicit realizations of class field theory for globalfunction fields, particularly through the theory of Hayes modules and the use of shtukas.They give rise to Galois representations whose image and ramification behavior encodesignificant arithmetic information. Moreover, the theory of heights and canonical measuresassociated with Drinfeld modules has led to important results in Diophantine geometry,such as analogues of the Mordell-Weil theorem and the Bogomolov conjecture in positivecharacteristic. Beyond their arithmetic significance, Drinfeld modules also exhibit a richdynamical structure.