SQUARENESS IN THE SPECIAL L-VALUE AND SPECIAL L-VALUES OF TWISTS

Let N be a prime and let A be a quotient of J0(N) over Q associated to a newform such that the special L-value of A (at s = 1) is non-zero. Suppose that the algebraic part of the special L-value of A is divisible by an odd prime q such that q does not divide the numerator of [Formula: see text]. Then the Birch and Swinnerton-Dyer conjecture predicts that the q-adic valuations of the algebraic part of the special L-value of A and of the order of the Shafarevich–Tate group are both positive even numbers. Under a certain mod q non-vanishing hypothesis on special L-values of twists of A, we show that the q-adic valuations of the algebraic part of the special L-value of A and of the Birch and Swinnerton-Dyer conjectural order of the Shafarevich–Tate group of A are both positive even numbers. We also give a formula for the algebraic part of the special L-value of A over quadratic imaginary fields K in terms of the free abelian group on isomorphism classes of supersingular elliptic curves in characteristic N (equivalently, over conjugacy classes of maximal orders in the definite quaternion algebra over Q ramified at N and ∞) which shows that this algebraic part is a perfect square up to powers of the prime two and of primes dividing the discriminant of K. Finally, for an optimal elliptic curve of arbitrary conductor E, we give a formula for the special L-value of the twist E-Dof E by a negative fundamental discriminant -D, which shows that this special L-value is an integer up to a power of 2, under some hypotheses. In view of the second part of the Birch and Swinnerton-Dyer conjecture, this leads us to the surprising conjecture that the square of the order of the torsion subgroup of E-Ddivides the product of the order of the Shafarevich–Tate group of E-Dand the orders of the arithmetic component groups of E-D, under certain mild hypotheses.

Several Diophantine Equations in Some Rings of Integers of Quadratic Imaginary Fields

In this paper, it is proved that the Diophantine equation x4-y4 =z2 has no non-trivial coprime solutions in the rings of integers of quadratic imaginary fields [Formula: see text] for d=11, 19, 43, 67, 163, which implies that the Fermat equation x4+y4 =z4 has no non-trivial solutions in these fields either. Then all the solutions of the Pocklington equation x4-x2y2+y4 =(-1)σz2 (σ =0 or 1) in the ring of integers of [Formula: see text] are determined, and as an application, the result is applied to K2 of a field.