p-adic and archimedean equidistribution of arithmetic cycles
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2024
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Abstract
The purpose of this manuscript is to describe various equidistribution problems related to arithmetic cycles. Depending on the case, the context is p-adic, archimedean or S-arithmetic. The starting point for this research is Duke's equidistribution theorem. Our first result is a p-adic analogue for this theorem in the case of closed modular geodesics. The techniques employed for this are strong enough to obtain other equidistribution results such as: equidistribution of CM points on the p-adic points of a Shimura curve ramified at p; equidistribution of ATR cycles on Hilbert varieties; equidistribution of Stark-Heegner cycles in the mock Hilbert modular surface. The first chapter is an Introduction for our motivations and main results. The next two chapters are devoted to the proof our theorems. There is also an Appendix deepening in the notion of p-adic geodesics.
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Tesis (Doctor en Matemáticas)--Pontificia Universidad Católica de Chile, 2024