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A lower bound for the rank of \(J_0(q)\). (English) Zbl 0973.11065

Let \(q\) be a prime number, \(J_0(q)\) the Jacobian variety of the modular curve \(X_0(N)\) over \(\mathbb Q\) and \(\dim(J_0(q))\sim q/12\). Eichler and Shimura [G. Shimura, Introduction to the Arithmetic Theory of Automorphic Forms, Princeton Univ. Press (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023)] showed that its \(L\)-function satisfies \(L(J_0(q),s)=\prod_{f\in S_2(q)^*}L(f,s)\), where \(S_2(q)^*\) denotes the space of primitive cusp forms of weight \(2\) and level \(q\), \(|S_2(q)^*|=\dim(J_0(q))\) and the \(L\)-functions are normalized so that \(\operatorname{Re}(s)=1/2\) is the critical line. The Birch and Swinnerton-Dyer conjecture would imply \(\text{rank}(J_0(q))=\sum_{f\in S_2(q)^*}\text{ord}_{s=1/2}L(f,s)\) and based on the heuristic of the zeros of \(L\)-functions it is expected \(\text{rank}(J_0(q))\sim \frac 12\dim(J_0(q))\). In a previous paper [E. Kowalski and P. Michel, Duke Math. J. 100, No. 3, 503-542 (1999; Zbl 1161.11359)], the authors used the previous decomposition of \(L(J_0(q),s)\) to obtain \(\text{rank}(J_0(q))\leq C\dim(J_0(q))\), for some absolute effectively computable constant \(C>0\) [E. Kowalski and P. Michel, Isr. J. Math. 120, Pt. A, 179-204 (2000; Zbl 0999.11026)].
In the present work, the authors are concerned with the dual problem of the non-vanishing of \(L(f,1/2)\), more precisely they look at forms \(f\) with order of vanishing exactly \(1\). Their first result is that for any \(\varepsilon>0\) and any \(q\) sufficiently large (in terms of \(\varepsilon\)) \[ \{f\in S_2(q)^*\mid L(f,1/2)=0,\;L'(f,1/2)\neq 0\}\geq \biggl(\frac{19}{54}-\varepsilon \biggr)|S_2(q)^*|. \] Using the work of B. Gross, W. Kohnen and D. Zagier [Math. Ann. 278, 497-562 (1987; Zbl 0641.14013)], the product \(\prod_fL(f,s)\) over the forms \(f\) with \(L(f,1/2)=0\) and \(L'(f,1/2)\neq 0\) is the \(L\)-function of a quotient of \(J_0(q)\) with rank exactly equal to the dimension. As a consequence for any \(\varepsilon>0\) and \(q\) large enough in terms of \(\varepsilon\), \[ \text{rank}(J_0(q))\geq \biggl(\frac{19}{54}-\varepsilon \biggr)\dim(J_0(q)). \] In fact this method works equally well (even more easily) for those forms \(f\) such that \(L(f,1/2)\neq 0\), i.e., they prove that for any \(\varepsilon>0\) and \(q\) large enough in terms of \(\varepsilon\) then \[ \{f\in S_2(q)^*\mid L(f,1/2)\neq 0\}\geq (\tfrac 16-\varepsilon)|S_2(q)^*|. \] However this result is weaker than that of H. Iwaniec and P. Sarnak [Isr. J. Math. 120, Pt. A, 155-177 (2000; Zbl 0992.11037)]. The main result of the paper is the following: for any \(0\leq\Delta<1/4\) and any \(q\) large enough in terms of \(\Delta\), they show \[ \sum_f1\geq\frac 12\left(1-\frac 1{(1+2\Delta)^3}\right)\dim(J_0(q)), \] where \(f\) runs through the forms in \(S_2(q)^*\) such that \(L(f,1/2)=0\) and \(L'(f,1/2)\neq 0\). The first result follows from this by letting \(\Delta\to 1/4\).

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
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