MODULAR FORM CONGRUENCES AND SELMER GROUPS

Motivated by Cremona and Mazurʹs notion of visibility of elements in Shafarevich–Tate groups [6, 27], there have been a number of recent works which test its compatibility with the Birch and Swinnerton–Dyer conjecture and the Bloch–Kato conjecture. These conjectures provide formulas for the orders of Shafarevich–Tate groups in terms of values of L-functions. For example, one may see recent work of Agashe, Dummigan, Stein and Watkins [1, 2, 10, 11]. In their examples, they find that the presence of visible elements agrees with the expected divisibility properties of the relevant L-values.