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The rank of $$J_ 0(N)$$. (English) Zbl 0851.11035
Columbia University number theory seminar, New York, 1992. Paris: Société Mathématique de France, Astérisque. 228, 41-68 (1995).
Let $$W(N)$$ be the Atkin-Lehner group acting on $$J_0(N)$$, and for each character $$\chi$$ on $$W(N)$$ let $$A_\chi (N)$$ be the corresponding subvariety of $$J_0(N)$$. Assume the Riemann hypothesis for $$L$$-functions of the new forms of level $$N$$. It is then shown that for any $$\varepsilon > 0$$ one has $\text{rank} A_\chi (N) \leq \left ({3 \over 2} + \varepsilon \right) \dim A_\chi (N),$ providing that $$N \geq N (\varepsilon)$$. Here rank $$A_\chi (N)$$ denotes the analytic rank.
Assuming also the Riemann hypothesis for Dirichlet $$L$$-functions with real characters, it is shown that $\sum_{Y < N \leq cY} \text{rank} J_0 (N) \leq \left( {7 \over 6} + \varepsilon \right) \sum_{Y < N \leq cY} \dim J_0 (N)$ for $$Y \geq Y (\varepsilon)$$, where $$N$$ runs over primes.
The proofs of these results use an explicit formula of Weil type, in the same way as in the author’s work [Invent. Math. 109, 445-472 (1992; Zbl 0783.14019)]. A new ingredient however is the Eichler-Selberg trace formula, which gives better estimates than the Hasse-Weil bound for small primes.
For the entire collection see [Zbl 0815.00008].

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 14H40 Jacobians, Prym varieties 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 11G18 Arithmetic aspects of modular and Shimura varieties 11F66 Langlands $$L$$-functions; one variable Dirichlet series and functional equations 14K15 Arithmetic ground fields for abelian varieties