## The rank of quotients of $$J_0(N)$$.(English)Zbl 1013.11030

From the introduction: The author extends the results of W. Duke [Invent. Math. 119, 165-174 (1995; Zbl 0838.11035)] on $$L$$-functions with low-order zeros at $$s = 1$$ with the following two (unconditional) theorems.
Theorem 1.1. There exists a positive constant $$a_1$$ such that for sufficiently large $$N$$, prime, there is a quotient of $$J_0(N)$$ with rank zero over $$\mathbb{Q}$$ that has dimension at least $$a_1D_N$$.
Theorem 1.2. There exists a positive constant $$b_1$$ such that for sufficiently large $$N$$, prime, the Jacobian of $$X^*_0(N) = X_0(N)/w_N$$ has rank at least $$b_1D_N$$ over $$\mathbb{Q}$$. Here $$w_N$$ denotes the Atkin-Lehner involution $$z\to -1/N_z$$.
He gives lower bounds for $$a_1$$ and $$b_1$$, but these are certainly far short of optimal. He proves the necessary results for a wider class of $$L$$-functions, namely, those twisted by Dirichlet characters.
He further proves the following, which (setting $$\chi =1$$) implies Theorem 1.1 through the work of Kolyvagin.
Theorem 1.3. Fix a character $$\chi$$. For $$N$$ large and prime, at least $$a_\chi D_N$$ of the $$L_f(1,\chi)$$’s are nonzero, where $$a_\chi > 0$$ depends only on $$\chi$$. In particular, for $$\chi =1$$, at least $$1/48-\varepsilon$$ are nonzero for sufficiently large $$N$$ depending on $$\varepsilon > 0$$.
Finally, he proves the following, which (for $$\chi = 1$$ and $$n = 1$$) implies Theorem 1.2 upon applying the results of Gross and Zagier.
Theorem 1.4. Fix $$\chi$$ a real character, and let $$n$$ be a positive integer. For $$N$$ large and prime, at least $$b^{(n)}_\chi$$ of the $$\xi_f(s,\chi)$$ have a nonzero $$n$$th derivative at $$s = 1$$, where $$b^{(n)}_\chi > 0$$ depends only on $$n$$ and $$\chi$$. In particular, one may take $$b^{(1)}_1$$ to be at least 0.015.
His methods follow those of Duke, in that he examines first and second moments of the $$\xi_f$$’s, but with two notable changes, each of which saves a factor of $$\log N$$ in the desired proportions. First, he uses the Eichler-Selberg trace formula for Hecke operators over $$S_2(\Gamma_0(N))$$, rather than Petersson’s formula for arithmetically weighted averages of eigenvalues of Hecke operators. Second, employing an idea dating back at least to Selberg, he weights the $$\xi_f$$’s with “mollifiers” $$m_f$$, designed to lessen the influence of the largest $$\xi_f$$’s on the sum.

### 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 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 11G18 Arithmetic aspects of modular and Shimura varieties

Zbl 0838.11035
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### References:

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