Berkovich Space

A Berkovich space is a non-Archimedean analytic space whose local charts include multiplicative seminorms as points, rather than only points with coordinates in finite extensions of the ground field. For an algebra of convergent power series over a complete non-Archimedean field , modulo an ideal, its Berkovich spectrum consists of bounded multiplicative seminorms extending the absolute value on . The topology is the weakest one making evaluation on each algebra element continuous (Conrad 2008).

For example, the Berkovich affine line consists of multiplicative seminorms on extending that absolute value. Besides evaluation at field-valued points, it contains disk seminorms such as

where and . These extra points allow analytic disks and their attachment points to encode connected geometry.