## 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

### Keywords:

modular curves; modular forms; $$L$$-functions; nonvanishing
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