William Stein's homepage

Tamagawa numbers

Theorem 3   When $p^2\nmid N$, the number c_p can be explicitly computed (up to a power of 2).

I prove this in [25]. Several related problems remain: when $p^2 \mid N$ it may be possible to compute c_p using the Drinfeld-Katz-Mazur interpretation of X₀(N); it should also be possible to use my methods to treat optimal quotients of J₁(N).

I was surprised to find that systematic computations using this formula indicate the following conjectural refinement of a result of Mazur [16].

Conjecture 4   Suppose N is prime and A is an optimal quotient of J₀(N). Then $A(\mathbf{Q})_{\tor}$ is generated by the image of $(0)-(\infty)$and $c_p = \char93 A(\mathbf{Q})_{\tor}$. Furthermore, the product of the c_p over all optimal factors equals the numerator of (N-1)/12.

I have checked this conjecture for all $N\leq 997$ and, up to a power of 2, for all $N\leq 2113$. The first part is known when A is an elliptic curve (see [20]). Upon hearing of this conjecture, Mazur proved it when all ``q-Eisenstein quotients'' are simple. There are three promising approaches to finding a complete proof. One involves the explicit formula of Theorem 3; another is based on Ribet's level lowering theorem, and a third makes use of a simplicity result of Merel.

Theorem 3 also suggests a way to compute Tamagawa numbers of motives attached to eigenforms of higher weight. These numbers appear in the conjectures of Bloch and Kato, which generalize the BSD conjecture to motives (see [5]).