A lower bound for the rank of J0(q)
نویسنده
چکیده
This paper is the second of a series devoted to the study of the rank of J0(q) (the Jacobian of the modular curve X0(q)), from the analytic point of view stemming from the Birch and Swinnerton-Dyer conjecture, which is tantamount to the study, on average, of the order of vanishing at the central critical point of the L-functions of primitive weight two forms f of level q (q prime). We prove that, for a large proportion of such forms, the associated L function vanishes at order exactly one at the critical point. From the work of Gross-Zagier, this implies a strong lower bound for the geometric rank of J0(q).
منابع مشابه
Explicit Upper Bound for the (Analytic) Rank of J0(q)
We refine the techniques of our previous paper [KM1] to prove that the average order of vanishing of L-functions of primitive automorphic forms of weight 2 and prime level q satisfies ∑ f∈S2(q)∗ ords= 1 2 L(f, s) ≤ C|S2(q) ∗| with C = 6.5, for all q large enough. On the Birch and Swinnerton-Dyer conjecture, this implies rank J0(q) ≤ C dim J0(q) for q prime large enough.
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