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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
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