Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups 6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins 1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.