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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

Citations:

Zbl 0838.11035
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References:

[1] Armand Brumer, The rank of \(J_ 0(N)\) , Astérisque (1995), no. 228, 3, 41-68, in Columbia University Number Theory Seminar (New York, 1992), Soc. Math. France, Montrouge. · Zbl 0851.11035
[2] D. A. Burgess, On character sums and \(L\)-series. II , Proc. London Math. Soc. (3) 13 (1963), 524-536. · Zbl 0123.04404
[3] Harold Davenport, Multiplicative number theory , Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 1980, 2d ed. · Zbl 0453.10002
[4] W. Duke, The critical order of vanishing of automorphic \(L\)-functions with large level , Invent. Math. 119 (1995), no. 1, 165-174. · Zbl 0838.11035
[5] B. Gross, W. Kohnen, and D. Zagier, Heegner points and derivatives of \(L\)-series. II , Math. Ann. 278 (1987), no. 1-4, 497-562. · Zbl 0641.14013
[6] Benedict H. Gross and Don B. Zagier, Heegner points and derivatives of \(L\)-series , Invent. Math. 84 (1986), no. 2, 225-320. · Zbl 0608.14019
[7] S. Kamienny, Torsion points on elliptic curves over fields of higher degree , Internat. Math. Res. Notices (1992), no. 6, 129-133. · Zbl 0807.14022
[8] V. Kolyvagin and D. Logachev, Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties , Leningrad Math. J. 1 (1990), 1229-1253. · Zbl 0728.14026
[9] B. Mazur, Modular curves and the Eisenstein ideal , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33-186 (1978). · Zbl 0394.14008
[10] Loïc Merel, Bornes pour la torsion des courbes elliptiques sur les corps de nombres , Invent. Math. 124 (1996), no. 1-3, 437-449. · Zbl 0936.11037
[11] Jean-François Mestre, Formules explicites et minorations de conducteurs de variétés algébriques , Compositio Math. 58 (1986), no. 2, 209-232. · Zbl 0607.14012
[12] M. Ram Murty, The analytic rank of \(J_ 0(N)(\mathbf Q)\) , Number theory (Halifax, NS, 1994), CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1995, pp. 263-277. · Zbl 0851.11036
[13] René Schoof and Marcel van der Vlugt, Hecke operators and the weight distributions of certain codes , J. Combin. Theory Ser. A 57 (1991), no. 2, 163-186. · Zbl 0729.11065
[14] A. Selberg, On the zeros of Riemann’s zeta-function , Skr. Norske Vid. Akad. Oslo I (1942), no. 10, reprinted in Collected Papers, Vol. I, Springer-Verlag, Berlin, 1989, 85-141. · Zbl 0028.11101
[15] Jean-Pierre Serre, Répartition asymptotique des valeurs propres de l’opérateur de Hecke \(T_ p\) , J. Amer. Math. Soc. 10 (1997), no. 1, 75-102. JSTOR: · Zbl 0871.11032
[16] Goro Shimura, Introduction to the arithmetic theory of automorphic functions , Publications of the Mathematical Society of Japan, No. 11. Iwanami Shoten, Publishers, Tokyo, 1971. · Zbl 0221.10029
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