# Roth's Theorem Over Adelic Curves

## Degeneracy of Algebraic Points April 24, 2023 - April 28, 2023

**Speaker(s):**Paul Vojta (University of California, Berkeley)

**Location:**SLMath: Eisenbud Auditorium, Online/Virtual

**Tags/Keywords**

arithmetic function fields

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**

#### Roth's Theorem Over Adelic Curves

Recently, P. Dolce and F. Zucconi proved that Roth's theorem holds over adelic curves, under certain hypotheses.

Here, an adelic curve is a field $K$, together with a collection of absolute values parameterized by a measure space, such that an analogue of the product formula holds for all elements of $K^{*}$. (This definition was formulated by H. Chen and A. Moriwaki, and encompasses number fields, function fields, and Moriwaki's concept of arithmetic function fields.)

We will briefly describe adelic curves and some examples, and then summarize the proof of the theorem of Dolce and Zucconi.

#### Roth's Theorem Over Adelic Curves

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