Birch and Swinnerton-Dyer Conjecture disproved – Counter examples and irregularities discovered for certain use cases of the Hasse Weil L Function

The Birch and Swinnerton-Dyer conjecture states that the rank of any given elliptic curve E should be equal to the order of zero of the Hasse Weil L function at point s = 1 of the same elliptic curve. This conjecture is developed by the mathematicians Bryan John Birch and Peter Swinnerton-Dyer in the 1960s with computer assistance. Although there exists some numerical evidence up to this day supporting this conjecture, particularly for lower ranked elliptic curves, the BSD conjecture has only been proven for some special cases like lower ranks and with some specific assumptions about the “underlying” elliptic curves. In this piece of work, we will analyze different valid cases applicable to the Hasse Weil L function and its infinite Euler product representation (which is a crucial part of the whole BSD conjecture), especially for the 2 cases of “good reduction” and “multiplicative reduction” that provides us with a good theoretical foundation for reevaluating the validity of the BSD conjecture. Finally, we are able to discover 2 counter examples, 1 for each of the 2 cases above, to the BSD conjecture which disproves the conjecture by showing that given these 2 cases above and under specific conditions, the rank of a given elliptic curve E on left-hand side (LHS) could never be equal to the order of zero of the Hasse Weil L function (at s = 1) of the same elliptic curve on the right-hand side (RHS) of the BSD conjecture statement.