×

Rankin-Selberg \(L\)-functions and the reduction of CM elliptic curves. (English) Zbl 1380.11060

Summary: Let \(q\) be a prime and \(K=\mathbb Q(\sqrt{-D})\) be an imaginary quadratic field such that \(q\) is inert in \(K\). If \(\mathfrak {q}\) is a prime above \(q\) in the Hilbert class field of \(K\), there is a reduction map \[ r_{\mathfrak q}: \mathcal{E}\ell \ell(\mathcal {O}_K) \longrightarrow {\mathcal {E}\ell \ell}^{ss}(\mathbb F_{q^2}) \] from the set of elliptic curves over \(\overline{\mathbb Q}\) with complex multiplication by the ring of integers \(\mathcal {O}_K\) to the set of supersingular elliptic curves over \(\mathbb {F}_{q^2}\). We prove a uniform asymptotic formula for the number of CM elliptic curves which reduce to a given supersingular elliptic curve and use this result to deduce that the reduction map is surjective for \(D \gg _{\varepsilon} q^{18+\varepsilon}\). This can be viewed as an analog of Linnik’s theorem on the least prime in an arithmetic progression. We also use related ideas to prove a uniform asymptotic formula for the average \[ \sum _{\chi }L(f \times \Theta _\chi ,1/2) \] of central values of the Rankin-Selberg \(L\)-functions \({L(f \times {\Theta _{\chi}},s)}\) where \(f\) is a fixed weight 2, level \(q\) arithmetically normalized Hecke cusp form and \(\Theta _\chi \) varies over the weight 1, level \(D\) theta series associated to an ideal class group character \(\chi \) of \(K\). We apply this result to study the arithmetic of Abelian varieties, subconvexity, and \(L^4\) norms of automorphic forms.

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11G05 Elliptic curves over global fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bertolini, M., Darmon, H.: A rigid analytic Gross-Zagier formula and arithmetic applications. Ann. Math. 146, 111-147 (1997) · Zbl 1029.11027 · doi:10.2307/2951833
[2] Blomer, V., Harcos, G.: Hybrid bounds for twisted L-functions. J. Reine Angew. Math. 621, 53-79 (2008) · Zbl 1193.11044
[3] Blomer, V., Khan, R., Young, M.: Distribution of mass of Hecke eigenforms. Duke Math. J. 162, 2609-2644 (2013) · Zbl 1312.11028 · doi:10.1215/00127094-2380967
[4] Blomer, V., Michel, P.: Sup-norms of eigenfunctions on arithmetic ellipsoids. Int. Math. Res. Not. IMRN 21, 4934-4966 (2011) · Zbl 1294.11075
[5] Buttcane, J., Khan, R.: L norms of Hecke newforms of large level. http://arxiv.org/abs/1305.1850 (2013, preprint) · Zbl 1336.11032
[6] Cornut, C.: Mazur’s conjecture on higher Heegner points. Invent. Math. 148, 495-523 (2002) · Zbl 1111.11029 · doi:10.1007/s002220100199
[7] Duke, W.: Hyperbolic distribution problems and half-integral weight Maass forms. Invent. Math. 92, 73-90 (1988) · Zbl 0628.10029 · doi:10.1007/BF01393993
[8] Duke, W.: The critical order of vanishing of automorphic L-functions with large level. Invent. Math. 119, 165-174 (1995) · Zbl 0838.11035 · doi:10.1007/BF01245178
[9] Elkies, N., Ono, K., Yang, T.H.: Reduction of CM elliptic curves and modular function congruences. Int. Math. Res. Not. 44, 2695-2707 (2005) · Zbl 1166.11323 · doi:10.1155/IMRN.2005.2695
[10] Feigon, B., Whitehouse, D.: Averages of central L-values of Hilbert modular forms with an application to subconvexity. Duke Math. J. 149, 347-410 (2009) · Zbl 1241.11057 · doi:10.1215/00127094-2009-041
[11] Feigon, B., Whitehouse, D.: Exact averages of central values of triple product L-functions. Int. J. Number Theory 6, 1609-1624 (2010) · Zbl 1236.11050 · doi:10.1142/S179304211000368X
[12] Gelbart, S., Jacquet, H.: A relation between automorphic representations of GL(2) and GL(3). Ann. Sci. École Norm. Sup. 11, 471-542 (1978) · Zbl 0406.10022
[13] Gross, B.: Heights and the special values of L-series. Number theory (Montreal, Que., 1985). In: CMS Conference Proceedings, vol. 7, pp. 115-187. Amer. Math. Soc., Providence (1987) · Zbl 0623.10019
[14] Gross, B., Kudla, S.: Heights and the central critical values of triple product L-functions. Compositio Math. 81, 143-209 (1992) · Zbl 0807.11027
[15] Hoffstein, J., Kontorovich, A.: The first non-vanishing quadratic twist of an automorphic L-series. http://arxiv.org/abs/1008.0839 (2010, preprint) · Zbl 0406.10022
[16] Holowinsky, R., Munshi, R.: Level aspect subconvexity for Rankin-Selberg L-functions, to appear in automorphic representations and L-functions. Tata Institute of Fundamental Research, Mumbai. http://arxiv.org/abs/1203.1300 (2012) · Zbl 1305.11036
[17] Holowinsky, R., Templier, N.: First moment of Rankin-Selberg central L-values and subconvexity in the level aspect. Ramanujan J. 33, 131-155 (2014) · Zbl 1290.11085 · doi:10.1007/s11139-012-9454-y
[18] Iwaniec, H.: Fourier coefficients of modular forms of half-integral weight. Invent. Math. 87, 385-401 (1987) · Zbl 0606.10017 · doi:10.1007/BF01389423
[19] Iwaniec, H., Kowalski, E.: Analytic number theory. American Mathematical Society Colloquium Publications, vol. 53, pp. xii+615. American Mathematical Society, Providence (2004) · Zbl 1059.11001
[20] Iwaniec, H., Luo, W., Sarnak, P.: Low lying zeros of families of L-functions. Inst. Hautes Études Sci. Publ. Math. 91(2000), 55-131 (2001) · Zbl 1012.11041
[21] Jackson, J., Knightly, A.: Averages of twisted L-functions, p. 24 (2015, preprint) · Zbl 1327.11034
[22] Jetchev, D., Kane, B.: Equidistribution of Heegner points and ternary quadratic forms. Math. Ann. 350, 501-532 (2011) · Zbl 1237.11026 · doi:10.1007/s00208-010-0568-5
[23] Kane, B.: Representations of integers by ternary quadratic forms. Int. J. Number Theory 6, 127-159 (2010) · Zbl 1250.11037 · doi:10.1142/S1793042110002831
[24] Kane, B.: CM liftings of supersingular elliptic curves. J. Théor. Nombres Bordeaux 21, 635-663 (2009) · Zbl 1214.11142 · doi:10.5802/jtnb.692
[25] Kohel, D.: Endomorphism rings of elliptic curves over finite fields. University of California, Berkeley, Thesis (1996) · Zbl 1312.11028
[26] Kohnen, W., Sengupta, J.: On quadratic character twists of Hecke L-functions attached to cusp forms of varying weights at the central point. Acta Arith. 99, 61-66 (2001) · Zbl 0982.11026 · doi:10.4064/aa99-1-5
[27] Kohnen, W., Zagier, D.: Values of L-series of modular forms at the center of critical strip. Invent. Math. 64, 175-198 (1981) · Zbl 0468.10015 · doi:10.1007/BF01389166
[28] Li, X.: Upper bounds on L-functions at the edge of the critical strip. IMRN 4, 727-755 (2010) · Zbl 1219.11136
[29] Liu, S.-C., Masri, R., Young, M.P.: Subconvexity and equidistribution of Heegner points in the level aspect. Compositio Math. 149, 1150-1174 (2013) · Zbl 1329.11046 · doi:10.1112/S0010437X13007033
[30] Michel, P.: The subconvexity problem for Rankin-Selberg L-functions and equidistribution of Heegner points. Ann. Math. 160, 185-236 (2004) · Zbl 1068.11033 · doi:10.4007/annals.2004.160.185
[31] Michel, P., Ramakrishnan, D.: Consequences of the Gross-Zagier formulae: stability of average L. Number theory, analysis and geometry, pp. 437-459. Springer, New York (2012) · Zbl 1276.11057
[32] Michel, P., Venkatesh, A.: Heegner points and non-vanishing of Rankin/Selberg L-functions. Analytic number theory, Clay Math. Proc., vol. 7, pp. 169-183. Amer. Math. Soc., Providence (2007) · Zbl 1214.11063
[33] Nelson, P.: Stable averages of central values of Rankin-Selberg L-functions: some new variants. http://arxiv.org/abs/1202.6313 (2012, preprint) · Zbl 1288.11050
[34] Vatsal, V.: Uniform distribution of Heegner points. Invent. Math. 148, 1-46 (2002) · Zbl 1119.11035 · doi:10.1007/s002220100183
[35] Yang, T.H.: Minimal CM liftings of supersingular elliptic curves. Pure Appl. Math. Q. 4, 1317-1326 (2008) · Zbl 1159.14015 · doi:10.4310/PAMQ.2008.v4.n4.a14
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.