##
**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.

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.

Reviewer: O.Ninnemann (Berlin)

### 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 |

### Keywords:

mollifiers; \(L\)-functions with low-order zeros; Eichler-Selberg trace formula; Hecke operators### Citations:

Zbl 0838.11035
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\textit{J. M. VanderKam}, Duke Math. J. 97, No. 3, 545--577 (1999; Zbl 1013.11030)

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### References:

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