On the Birch-Swinnerton-Dyer conjectural formula for certain quotients of J_{0}(N)

Amod AGASHÉ Department of Mathematics, 940 Evans Hall, University of California, Berkeley, CA 94720, U.S.A. Email: amod@math.berkeley.edu

Abstract. This is an outline of my U.C. Berkeley Mathematics Ph.D. thesis.

1 The Birch-Swinnerton-Dyer conjectural formula for rank

zero quotients of J_{0}(N)

Definitions, statement, etc. eg from §2 of [Ag-inv] (nothing new):

\frac{L(A, 1)}{Ω(A)} =^{?} \frac{|� _{A}| ∏_{p|N} c_{p}}{|A(Q)_{tor}||Â(Q)_{tor}|}

2 A generalization of the Manin constant and the relation

between congruence primes and modular degree

[Ag-man], also write out the proof of the latter using K.Ribet’s ideas (multiplicity one proofs).

3 A formula for the ratio of the special L-value to the real

period for the winding quotient of prime level

§2 of [Ag-inv]:

\frac{L(J_{e}, 1)}{Ω(J_{e})} = \frac{1}{c_{J_{e}}} \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}}||\frac{H_{e}^{+}}{ℑe}|}{n}

4 Comparison of the results of section 3 to the conjecture

in section 1

• Calculate the order of the torsion group of J_{e} and its dual using a result due to J.Oesterle and another due to L.Merel repectively.

• Relate the first factor | \frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}} | in Prop. 2.1 of [Ag-inv] to congruences.

• Now compare to reduce the BSD conjecture to:

\frac{|� _{J_{e}}| c_{p}}{n^{2}} =^{?} \frac{1}{c_{J_{e}}} \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}}||\frac{H_{e}^{+}}{ℑe}|}{n}.

(The rest is speculative:)

• Is c_{p} Eisenstein? Then one may expect c_{p} = n.

Assuming this, and assuming c_{J_{e}} = 1, the conjecture reduces to:

|� _{J_{e}}| =^{?} |\frac{H^{+}}{Ĥ_{e}^{+} + H_{e}^{+}} | |\frac{H_{e}^{+}}{ℑe} | .  (1)

• Use B. Mazur’s ideas of constructing element of the Tate-Shafarevich group using congruences between modular forms with different analytic rank to show that |\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}}| should divide |� _{J_{e}}|.

• That leaves the factor | \frac{H_{e}^{+}}{ℑe} |: perhaps it measures congruences at higher level and hence may divide |� _{J_{e}}|.

• Thus one may be able to show that the order of the “analytic � ” divides the order of the (algebraic) � . Then results of Kolyvagin and Kato give results in the other direction, and this “may” prove the conjecture (eg in particular cases).

5 A formula for the ratio of the special L-value to the real period for quotients associated to newforms

§2.7 from [AgSt], but written up differently (to be done):

\frac{L(A_{f}, 1)}{Ω(A_{f})} = \frac{1}{c_{A_{f}}} [H_{0}(A_{f}, Z) : Te/I_{f}e]

6 A formula for the ratio of the special L-value to the real period for the winding quotient of level a product of two distinct primes

[Ag-pq]:

\frac{L(J_{e}, 1)}{Ω(J_{e})} = \frac{|\frac{H^{+}}{Ĥ_{e}^{+}+H_{e}^{+}}||\frac{H_{e}^{+}}{ℑe}|}{∏_{f∈S_{p}}(1 + q − a_{q}(f)) ∏_{g∈S_{q}}(1 + p − a_{p}(g)) |Te/ℑe|} \frac{∏_{f∈S_{p}} q ∏_{g∈S_{q}} p}{c_{J_{e}}}

7 Detecting invisible elements using “the modular degree”

§3 from [Ag-inv] (generalization of ideas of B. Mazur) and §4 of [Ag-inv].

8 Calculations of certain factors from section 3

[Ag-calc], algorithms and tables of calculations (all done, but to be written up)

References

[Ag-inv] Agashe, A.: On the ratio of the L-value to the real period for winding quotients of prime level and the detection of invisible elements of its Tate-Shafarevich group, preprint, being an expanded version of: Agashe, A.: On invisible elements of the Tate-Shafarevich group. C. R. Acad. Sci. Paris Ser. I Math. 328 (1999), no. 5, 369–374.

[Ag-calc] Agashe, A.: On the calculation of | \frac{H_{e}^{+}}{ℑe} |, preprint. [Ag-pq] Agashe A., On the ratio of the special L-value to the real period when the level is a product

of two distinct primes, preprint. [Ag-man] Agashe A., A generalization of the Manin constant and the relationship between con-

gruence primes and modular degree, preprint.

[AgSt] Agashe A., Stein W., Some Abelian Varieties with Visible Shafarevich-Tate Groups, preprint.