Arithmetic and Geometry presents highlights of recent work in
arithmetic algebraic geometry by some of the world's leading
mathematicians. Together, these 2016 lectures--which were delivered in
celebration of the tenth anniversary of the annual summer workshops in
Alpbach, Austria--provide an introduction to high-level research on
three topics: Shimura varieties, hyperelliptic continued fractions and
generalized Jacobians, and Faltings height and L-functions. The book
consists of notes, written by young researchers, on three sets of
lectures or minicourses given at Alpbach.
The first course, taught by Peter Scholze, contains his recent results
dealing with the local Langlands conjecture. The fundamental question is
whether for a given datum there exists a so-called local Shimura
variety. In some cases, they exist in the category of rigid analytic
spaces; in others, one has to use Scholze's perfectoid spaces.
The second course, taught by Umberto Zannier, addresses the famous Pell
equation--not in the classical setting but rather with the so-called
polynomial Pell equation, where the integers are replaced by polynomials
in one variable with complex coefficients, which leads to the study of
hyperelliptic continued fractions and generalized Jacobians.
The third course, taught by Shou-Wu Zhang, originates in the
Chowla-Selberg formula, which was taken up by Gross and Zagier to relate
values of the L-function for elliptic curves with the height of Heegner
points on the curves. Zhang, X. Yuan, and Wei Zhang prove the
Gross-Zagier formula on Shimura curves and verify the Colmez conjecture
on average.
