Algorithm for determining the type of a singular fiber in an elliptic pencil

On the conjecture of Birch and Swinnerton-Dyer

“…To see that the last two types cannot occur, simply note that E 0 (K p ) isp-divisible (cf. [25]) but E(K p ) is not, by the analogue of Lemma 2.5(ii) forp.…”

At Most 64 Lines on Smooth Quartic Surfaces (Characteristic 2)

“…Then minimalizing is achieved by running Tate's algorithm [18] which consequently gives relations between the coefficients of a 1 , D ′ and ∆, or in some cases like ours leads to a contradiction. By inspection of (6), the polynomial a 1 encodes singular or supersingular fibers.…”

On quartics with lines of the second kind

“…It is trivial, but crucial to note that if X is smooth, the fibers of π are the residual cubics, so they consist of at most 3 components. By the classification of Kodaira [4] and Tate [9], this allows for six different types of singular fibers, listed below with corresponding vanishing order v of ∆:…”