The analytic rank of \(J_0(q)\) and zeros of automorphic \(L\)-functions.

*(English)*Zbl 1161.11359For this important paper we cite the introduction:

This paper is motivated by the conjecture of Birch and Swinnerton-Dyer relating the rank of the Mordell-Weil group of an abelian variety defined over a number field with (in its crudest form) the order of vanishing of its Hasse-Weil \(L\)-function at the central critical point. J.-F. Mestre [Compos. Math. 58, 209–232 (1986; Zbl 0607.14012)] began the study of the implications of this conjecture towards providing upper bounds for the rank. He used “explicit formulae” similar to that of Riemann-Weil and assumed the analytic continuation and (perhaps more significantly) the Riemann hypothesis for those \(L\)-functions.

A. Brumer [Astérisque 228, 41–68 (1995; Zbl 0851.11035)] first studied the special case of the Jacobian variety \(J_0(q)\) of the modular curve \(X_0(q)\). This is an abelian variety defined over \(Q\) of dimension about \(q/12\). Here analytic continuation is known, by the work of Eichler and G. Shimura [Introduction to the arithmetic theory of automorphic functions (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023)]. Assuming only the Riemann hypothesis for the \(L\)-functions of automorphic forms (of weight 2 and level \(q\)), Brumer proved

\[ \text{rank}_a J_0(q)\leq\left(\frac 32 +o(1)\right) \dim J_0(q) \]

and conjectured that \[ \text{rank} J_0(q) = \text{rank}_a J_0(q) \text{sym}\frac 12\dim J_0(q) \] (based on the fact that the sign of the functional equation for the automorphic \(L\)-functions of weight 2 and level \(q\) is approximately half the time \(+1\) and half the time \(-1\)).

Other authors, notably M. Ram Murty [CMS Conf. Proc. 15, 263–277 (1995; Zbl 0851.11036)] (who first applied the Petersson formula in this context), considered the same problem. Most recently, W. Luo, H. Iwaniec, and P. Sarnak [Low lying zeros for families of \(L\)-functions, preprint, 1998], using the same assumptions, proved an estimate \[ \text{rank}_a J_0(q) \leq (c+o(1))\dim J_0(q) \] for some (explicit) constant \(c < 1\). This turns out to be quite significant in light of the general conjectures of N. M. Katz and P. Sarnak [Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications. 45. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0958.11004)] on the distribution of zeros of families of \(L\)-functions.

This paper approaches the same problem with a different emphasis: we wish to avoid all assumptions about the \(L\)-functions involved and obtain a bound of the correct order of magnitude. Indeed, we prove the following theorem.

Theorem 1. There exists an absolute and effective constant \(C > 0\) such that for any prime number \(q\), \[ \text{rank}_a\;J_0 (q) \leq C \dim J_0 (q). \]

If the Birch and Swinnerton-Dyer conjecture holds for \(J_0 (q)\), then \[ \text{rank}_a\;J_0 (q) \leq C \dim J_0 (q). \]

This theorem provides the first known unconditional bound for the analytic rank of a family of \(L\)-functions that is of the correct order of magnitude, without using the generalized Riemann hypothesis. We were inspired by the unconditional bounds for the analytic rank of twists of elliptic curves obtained by J. Pomykała and A. Perelli [Acta Arith. 80, No. 2, 149–163; (1997; Zbl 0878.11022)].

No remotely comparable upper bound for rank \(J_0 (q)\) seems to be accessible by algebraic means today. The starting point of this work is the factorization of the Hasse-Weil zeta function of \(J_0 (q)\) due to Eichler and Shimura (completed by Carayol at the bad primes): \[ L(J_0 (q), s) = \prod_{S_2(q)^*} L(f,s+1/2) \tag{1} \] where \(f\) ranges over the finite set \(S_2(q)^*\) of primitive forms (newforms) \(f\) of weight 2 and level \(q\), and \(L(f, s)\) is the corresponding Hecke \(L\)-function normalized so that the critical line is \(\text{Re}(s) = 1/2\). Hence the order of vanishing of the \(L\)-function of \(J_0 (q)\) at \(s = 1/2\) is the sum of the order of vanishing of the Hecke \(L\)-functions at \(s = 1/2\), \[ \text{rank}_a\;J_0 (q) = \sum_{f \in S_2(q)^*}\text{ord}_{s=1/2} L(f, s) \]

and if the Birch and Swinnerton-Dyer conjecture holds, then

\[ \text{rank}\;J_0 (q) = \sum_{f \in S_2(q)^*}\text{ord}_{s=1/2} L(f, s) \]

Thus our main theorem is equivalent with the following theorem.

Theorem 2. There exists an absolute and effective constant \(C > 0\) such that for any prime number \(q\), we have \[ \sum_{f \in S_2(q)^*}\text{ord}_{s=1/2} L(f, s)\leq C | S_2(q)^*|. \]

The strategy that we use is based on the explicit formula, except that a much tighter control of the possible zeros outside the critical line is required. This is obtained by means of the density Theorem 4 for zeros of automorphic \(L\)-functions with imaginary parts as close as \(1/(\log q)\), which is the crucial scale in this problem. This density theorem is similar to one proved by A. Selberg [Skr. Norske Vid.-Akad., Oslo, I 1946, No. 3, 1–62 (1946; Zbl 0061.08404)] for Dirichlet characters and is based on the study of a mollified second moment of values of the \(L\)-functions close to the critical line (see below for details).

This proof is carried out in Sections 4 and 5 after some important preliminary results in Section 3. While we were working on this result, it was suggested to us by Iwaniec that the fundamental estimate in the proof of Theorem 1 (for a mollified second moment of values of the \(L\)-functions) can also be adapted to prove nonvanishing results for the special values \(L(f, 1/2)\), analogous to part of H. Iwaniec, and P. Sarnak [The nonvanishing of central values of automorphic \(L\)-functions and Siegel’s zeros, preprint, 1997 [IS]]. Specifically, this yields the following theorem.

Theorem 3. For any \(\varepsilon > 0\) and any \(q\) prime large enough (in terms of \(\varepsilon\)), we have

\[ |\{f \in S_2 (q)^* \mid L(f, 1/2)\not =0 \}|\geq (1/6-\varepsilon) S_2 (q)^*. \]

In [Zbl 0973.11065], the case of the special values of the derivatives \(L(f, 1/2)\), for form with \(L(f, 1/2) = 0\), is treated by the same method. We refer to this paper for some more details.

Duke [Zbl 0838.11035] proved that the number of forms \(f\) with \(L(f, 1/2) = 0\) was at least a positive multiple of \(q/(\log q)^2\) . Independently, J. M. Vanderkam [Duke Math. J. 97, No. 3, 545–577 (1999; Zbl 1013.11030)] proved that there is a positive proportion (although with smaller constant) of forms with \(L(f, 1/2) = 0\).

This provides a lower bound for the dimension of the winding quotient of L. Merel [Invent. Math. 124, No. 1–3, 437–449 (1996; Zbl 0936.11037)]. In particular, the work of V. A. Kolyvagin and D. Y. Logachev [Leningr. Math. J. 1, 1229–1253 (1990; Zbl 0728.14026)] implies that there is a quotient of \(J_0 (q)\) defined over \(\mathbb Q\) with finite Mordell-Weil group and dimension greater than or equal to \((1/6 + o(1)) \dim J_0 (q)\).

The work of [IS] contains, among many other results for various families of \(L\)-functions, a proof of the much more difficult fact that 1/6 can be replaced by 1/4. Moreover, 1/4 is the natural barrier in this problem in the sense (the original motivation of [IS]) that any constant greater than 1/4 (with some additional lower bound on \(L(f, 1/2)\), which is proved in [IS] to hold for 1/4) would prove that Landau-Siegel zeros do not exist for Dirichlet L-functions of quadratic characters or, equivalently, provide an effective lower bound for class numbers of imaginary quadratic fields \[ h(\mathbb Q(\sqrt{-D}))\gg\frac{\sqrt D}{(\log D)^2} \] for \(D > 0.\)

This paper is motivated by the conjecture of Birch and Swinnerton-Dyer relating the rank of the Mordell-Weil group of an abelian variety defined over a number field with (in its crudest form) the order of vanishing of its Hasse-Weil \(L\)-function at the central critical point. J.-F. Mestre [Compos. Math. 58, 209–232 (1986; Zbl 0607.14012)] began the study of the implications of this conjecture towards providing upper bounds for the rank. He used “explicit formulae” similar to that of Riemann-Weil and assumed the analytic continuation and (perhaps more significantly) the Riemann hypothesis for those \(L\)-functions.

A. Brumer [Astérisque 228, 41–68 (1995; Zbl 0851.11035)] first studied the special case of the Jacobian variety \(J_0(q)\) of the modular curve \(X_0(q)\). This is an abelian variety defined over \(Q\) of dimension about \(q/12\). Here analytic continuation is known, by the work of Eichler and G. Shimura [Introduction to the arithmetic theory of automorphic functions (1971; Zbl 0221.10029), reprint (1994; Zbl 0872.11023)]. Assuming only the Riemann hypothesis for the \(L\)-functions of automorphic forms (of weight 2 and level \(q\)), Brumer proved

\[ \text{rank}_a J_0(q)\leq\left(\frac 32 +o(1)\right) \dim J_0(q) \]

and conjectured that \[ \text{rank} J_0(q) = \text{rank}_a J_0(q) \text{sym}\frac 12\dim J_0(q) \] (based on the fact that the sign of the functional equation for the automorphic \(L\)-functions of weight 2 and level \(q\) is approximately half the time \(+1\) and half the time \(-1\)).

Other authors, notably M. Ram Murty [CMS Conf. Proc. 15, 263–277 (1995; Zbl 0851.11036)] (who first applied the Petersson formula in this context), considered the same problem. Most recently, W. Luo, H. Iwaniec, and P. Sarnak [Low lying zeros for families of \(L\)-functions, preprint, 1998], using the same assumptions, proved an estimate \[ \text{rank}_a J_0(q) \leq (c+o(1))\dim J_0(q) \] for some (explicit) constant \(c < 1\). This turns out to be quite significant in light of the general conjectures of N. M. Katz and P. Sarnak [Random matrices, Frobenius eigenvalues, and monodromy, Colloquium Publications. 45. Providence, RI: American Mathematical Society (AMS) (1999; Zbl 0958.11004)] on the distribution of zeros of families of \(L\)-functions.

This paper approaches the same problem with a different emphasis: we wish to avoid all assumptions about the \(L\)-functions involved and obtain a bound of the correct order of magnitude. Indeed, we prove the following theorem.

Theorem 1. There exists an absolute and effective constant \(C > 0\) such that for any prime number \(q\), \[ \text{rank}_a\;J_0 (q) \leq C \dim J_0 (q). \]

If the Birch and Swinnerton-Dyer conjecture holds for \(J_0 (q)\), then \[ \text{rank}_a\;J_0 (q) \leq C \dim J_0 (q). \]

This theorem provides the first known unconditional bound for the analytic rank of a family of \(L\)-functions that is of the correct order of magnitude, without using the generalized Riemann hypothesis. We were inspired by the unconditional bounds for the analytic rank of twists of elliptic curves obtained by J. Pomykała and A. Perelli [Acta Arith. 80, No. 2, 149–163; (1997; Zbl 0878.11022)].

No remotely comparable upper bound for rank \(J_0 (q)\) seems to be accessible by algebraic means today. The starting point of this work is the factorization of the Hasse-Weil zeta function of \(J_0 (q)\) due to Eichler and Shimura (completed by Carayol at the bad primes): \[ L(J_0 (q), s) = \prod_{S_2(q)^*} L(f,s+1/2) \tag{1} \] where \(f\) ranges over the finite set \(S_2(q)^*\) of primitive forms (newforms) \(f\) of weight 2 and level \(q\), and \(L(f, s)\) is the corresponding Hecke \(L\)-function normalized so that the critical line is \(\text{Re}(s) = 1/2\). Hence the order of vanishing of the \(L\)-function of \(J_0 (q)\) at \(s = 1/2\) is the sum of the order of vanishing of the Hecke \(L\)-functions at \(s = 1/2\), \[ \text{rank}_a\;J_0 (q) = \sum_{f \in S_2(q)^*}\text{ord}_{s=1/2} L(f, s) \]

and if the Birch and Swinnerton-Dyer conjecture holds, then

\[ \text{rank}\;J_0 (q) = \sum_{f \in S_2(q)^*}\text{ord}_{s=1/2} L(f, s) \]

Thus our main theorem is equivalent with the following theorem.

Theorem 2. There exists an absolute and effective constant \(C > 0\) such that for any prime number \(q\), we have \[ \sum_{f \in S_2(q)^*}\text{ord}_{s=1/2} L(f, s)\leq C | S_2(q)^*|. \]

The strategy that we use is based on the explicit formula, except that a much tighter control of the possible zeros outside the critical line is required. This is obtained by means of the density Theorem 4 for zeros of automorphic \(L\)-functions with imaginary parts as close as \(1/(\log q)\), which is the crucial scale in this problem. This density theorem is similar to one proved by A. Selberg [Skr. Norske Vid.-Akad., Oslo, I 1946, No. 3, 1–62 (1946; Zbl 0061.08404)] for Dirichlet characters and is based on the study of a mollified second moment of values of the \(L\)-functions close to the critical line (see below for details).

This proof is carried out in Sections 4 and 5 after some important preliminary results in Section 3. While we were working on this result, it was suggested to us by Iwaniec that the fundamental estimate in the proof of Theorem 1 (for a mollified second moment of values of the \(L\)-functions) can also be adapted to prove nonvanishing results for the special values \(L(f, 1/2)\), analogous to part of H. Iwaniec, and P. Sarnak [The nonvanishing of central values of automorphic \(L\)-functions and Siegel’s zeros, preprint, 1997 [IS]]. Specifically, this yields the following theorem.

Theorem 3. For any \(\varepsilon > 0\) and any \(q\) prime large enough (in terms of \(\varepsilon\)), we have

\[ |\{f \in S_2 (q)^* \mid L(f, 1/2)\not =0 \}|\geq (1/6-\varepsilon) S_2 (q)^*. \]

In [Zbl 0973.11065], the case of the special values of the derivatives \(L(f, 1/2)\), for form with \(L(f, 1/2) = 0\), is treated by the same method. We refer to this paper for some more details.

Duke [Zbl 0838.11035] proved that the number of forms \(f\) with \(L(f, 1/2) = 0\) was at least a positive multiple of \(q/(\log q)^2\) . Independently, J. M. Vanderkam [Duke Math. J. 97, No. 3, 545–577 (1999; Zbl 1013.11030)] proved that there is a positive proportion (although with smaller constant) of forms with \(L(f, 1/2) = 0\).

This provides a lower bound for the dimension of the winding quotient of L. Merel [Invent. Math. 124, No. 1–3, 437–449 (1996; Zbl 0936.11037)]. In particular, the work of V. A. Kolyvagin and D. Y. Logachev [Leningr. Math. J. 1, 1229–1253 (1990; Zbl 0728.14026)] implies that there is a quotient of \(J_0 (q)\) defined over \(\mathbb Q\) with finite Mordell-Weil group and dimension greater than or equal to \((1/6 + o(1)) \dim J_0 (q)\).

The work of [IS] contains, among many other results for various families of \(L\)-functions, a proof of the much more difficult fact that 1/6 can be replaced by 1/4. Moreover, 1/4 is the natural barrier in this problem in the sense (the original motivation of [IS]) that any constant greater than 1/4 (with some additional lower bound on \(L(f, 1/2)\), which is proved in [IS] to hold for 1/4) would prove that Landau-Siegel zeros do not exist for Dirichlet L-functions of quadratic characters or, equivalently, provide an effective lower bound for class numbers of imaginary quadratic fields \[ h(\mathbb Q(\sqrt{-D}))\gg\frac{\sqrt D}{(\log D)^2} \] for \(D > 0.\)

Reviewer: Olaf 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 |

11G10 | Abelian varieties of dimension \(> 1\) |

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\textit{E. Kowalski} and \textit{P. Michel}, Duke Math. J. 100, No. 3, 503--542 (1999; Zbl 1161.11359)

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##### References:

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