| Thu, Apr 09, 2026
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Number Theory Seminar
3:15 PM
MSCS 509 | | Adelic Heights
Preston Kelley, OSU
Doctoral thesis defense. Note the special time.
| | | Abstract: A height function is a real-valued function which measures the complexity of a mathematical object. Height functions have played a critical role in areas such as diophantine geometry and arithmetic dynamics. In 2006, Favre and Rivera-Letelier introduced adelic heights, which form a large class of height functions on the space of algebraic numbers that can be studied using tools from potential theory. Although Favre and Rivera-Letelier's main application was proving a dynamical equidistribution theorem, there are also many interesting examples of adelic heights outside the context of dynamics. In this talk, I will discuss some of my work regarding adelic heights. In particular, I will discuss two classes of adelic heights. First, I will discuss adelic heights associated to sequences of rational maps. Heights associated to sequences of rational maps were previously studied by Kawaguchi in 2007, and they generalize dynamical heights. Our main result for these heights is that under suitable conditions, they can be realized as adelic heights. As a consequence, an equidistribution theorem holds. Second, I will discuss areal Weil heights, which are a class of adelic heights that generalize the areal Mahler measure, a variation of the Mahler measure introduced by Pritsker in 2008. I will present my main results for areal Weil heights, which include an analog of Kronecker's theorem, an equidistribution theorem, and a result demonstrating that an analog of Lehmer's Conjecture is false for certain areal Weil heights. I will conclude by discussing my ideas for future research on adelic heights. |
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