Extremal K3 surfaces {#ssec2.3}
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By the Shioda–Tate formula [@Shiodamodular; @Tate1; @Tate2], the following equality holds for an elliptic surface $S$ with a global section: $${\rm rk} ADE + {\rm rk} MW +2 = \rho(S).$$ We have used rk $ADE$ to denote the rank of the singularity type of an elliptic surface $S$. The Picard number $\rho(S)$ ranges from 2 to 20 for an elliptic K3 surface with a section. Thus, the rank of the singularity of an elliptic K3 surface $S$ with a section is bounded by: $${\rm rk} ADE = \rho(S)-2-{\rm rk} MW \le 18-{\rm rk} MW.$$ Therefore, the rank of the singularity of an elliptic K3 surface $S$ with a section can be 18 at the highest, and this value is achieved precisely when the Picard number attains the highest value 20, and the Mordell–Weil rank is 0. Physically, this means that the rank of the gauge group on F-theory compactification on an elliptic K3 surface is at most 18, and when the non-Abelian gauge group has the rank 18, it does not have a $U(1)$ gauge field.

K3 surfaces with Picard number 20 are called [*attractive*]{} K3 surfaces[^8]. The complex structure moduli of the attractive K3 