×

Canonical periods and congruence formulae. (English) Zbl 0979.11027

The results in this paper state roughly that congruences between the Fourier coefficients of modular forms give rise to congruences between the critical values for their associated \(L\)-functions. The author shows that periods may be selected for the associated \(L\)-functions so that one may precisely formulate the notion of a congruence between critical values. More precisely, the author proves the following. Let \(p\) be any odd prime and let \(f=\sum a_n q^n\) and \(g=\sum b_nq^n\) be cuspidal newforms of weight 2 on \(\Gamma_1(M)\), such that \(a_n\equiv b_n\pmod{\mathfrak p^r}\), for some prime \(\mathfrak p\) above \(p\) in \(\overline{\mathbb Q}\). Assume that \((M,p)=1\) and that the residual representation attached to \(f\) is irreducible. Fix an isomorphism \(\mathbb C_p \cong \mathbb C\), such that the prime \(p\) of \(\overline{\mathbb Q} \subset \mathbb C\) induces the usual absolute value on \(\mathbb C_p\). Then there exist canonical periods \(\Omega^{\pm}_f\) and \(\Omega^{\pm}_g\) such that the congruence \[ \tau(\overline{\chi}) \frac{L(1,f,\chi)}{-2\pi i \Omega^{\pm}_f} \equiv \tau(\overline{\chi}) \frac{L(1,g,\chi)}{-2\pi i \Omega^{\pm}_g} \pmod{\mathfrak p^r} \] holds for every character \(\chi\). Furthermore, there exists a \(\chi\) such that both sides of the congruence are nonzero modulo \(\mathfrak p\). The author also has some results in the residually reducible case as well.
As a corollary, the author shows that most elliptic curves \(E\) with a rational point of order 3 have the property that a positive proportion of its quadratic twists \(E_D\) have rank zero. He does this by employing a theorem of Davenport and Heilbronn on the distribution of 3-torsion in the class groups of quadratic number fields. These results generalize previous examples by W. Kohnen [J. Reine Angew. Math. 508, 179-187 (1999; Zbl 0968.11023)] and the reviewer [J. Am. Math. Soc. 11, 635-641 (1998; Zbl 0904.11015)].

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F30 Fourier coefficients of automorphic forms
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Software:

ecdata
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Avner Ash and Glenn Stevens, Modular forms in characteristic \(l\) and special values of their \(L\)-functions , Duke Math. J. 53 (1986), no. 3, 849-868. · Zbl 0618.10026 · doi:10.1215/S0012-7094-86-05346-9
[2] Spencer Bloch and Kazuya Kato, \(L\)-functions and Tamagawa numbers of motives , The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 333-400. · Zbl 0768.14001
[3] Henri Carayol, Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert , Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409-468. · Zbl 0616.10025
[4] John Coates, Motivic \(p\)-adic \(L\)-functions , \(L\)-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Ser., vol. 153, Cambridge Univ. Press, Cambridge, 1991, pp. 141-172. · Zbl 0725.11029 · doi:10.1017/CBO9780511526053.006
[5] J. E. Cremona, Algorithms for modular elliptic curves , Cambridge University Press, Cambridge, 1992. · Zbl 0758.14042
[6] H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields. II , Proc. Roy. Soc. London Ser. A 322 (1971), no. 1551, 405-420. · Zbl 0212.08101 · doi:10.1098/rspa.1971.0075
[7] Fred Diamond and John Im, Modular forms and modular curves , Seminar on Fermat’s Last Theorem (Toronto, ON, 1993-1994), CMS Conf. Proc., vol. 17, Amer. Math. Soc., Providence, RI, 1995, pp. 39-133. · Zbl 0853.11032
[8] Gerd Faltings and Bruce W. Jordan, Crystalline cohomology and \({\mathrm GL}(2,{\mathbf Q})\) , Israel J. Math. 90 (1995), no. 1-3, 1-66. · Zbl 0854.14010 · doi:10.1007/BF02783205
[9] G. Frey, On the Selmer group of twists of elliptic curves with \({\mathbf Q}\)-rational torsion points , Canad. J. Math. 40 (1988), no. 3, 649-665. · Zbl 0646.14024 · doi:10.4153/CJM-1988-028-9
[10] Dorian Goldfeld, Conjectures on elliptic curves over quadratic fields , Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979), Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 108-118. · Zbl 0417.14031
[11] Ralph Greenberg, Iwasawa theory for \(p\)-adic representations , Algebraic number theory, Adv. Stud. Pure Math., vol. 17, Academic Press, Boston, MA, 1989, pp. 97-137. · Zbl 0739.11045
[12] Ralph Greenberg and Glenn Stevens, \(p\)-adic \(L\)-functions and \(p\)-adic periods of modular forms , Invent. Math. 111 (1993), no. 2, 407-447. · Zbl 0778.11034 · doi:10.1007/BF01231294
[13] Haruzo Hida, Congruence of cusp forms and special values of their zeta functions , Invent. Math. 63 (1981), no. 2, 225-261. · Zbl 0459.10018 · doi:10.1007/BF01393877
[14] Haruzo Hida, Galois representations into \({\mathrm GL}_ 2({\mathbf Z}_ p[[X]])\) attached to ordinary cusp forms , Invent. Math. 85 (1986), no. 3, 545-613. · Zbl 0612.10021 · doi:10.1007/BF01390329
[15] Henryk Iwaniec, On the order of vanishing of modular \(L\)-functions at the critical point , Sém. Théor. Nombres Bordeaux (2) 2 (1990), no. 2, 365-376. · Zbl 0719.11029 · doi:10.5802/jtnb.33
[16] Kevin James, \(L\)-series with nonzero central critical value , J. Amer. Math. Soc. 11 (1998), no. 3, 635-641. JSTOR: · Zbl 0904.11015 · doi:10.1090/S0894-0347-98-00263-X
[17] W. Kohnen, On the proportion of quadratic twists of modular forms nonvanishing at the central critical point , preprint, 1997.
[18] B. Mazur, Modular curves and the Eisenstein ideal , Inst. Hautes Études Sci. Publ. Math. (1977), no. 47, 33-186 (1978). · Zbl 0394.14008 · doi:10.1007/BF02684339
[19] B. Mazur, On the arithmetic of special values of \(L\) functions , Invent. Math. 55 (1979), no. 3, 207-240. · Zbl 0426.14009 · doi:10.1007/BF01406841
[20] B. Mazur, J. Tate, and J. Teitelbaum, On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer , Invent. Math. 84 (1986), no. 1, 1-48. · Zbl 0699.14028 · doi:10.1007/BF01388731
[21] B. Mazur and A. Wiles, On \(p\)-adic analytic families of Galois representations , Compositio Math. 59 (1986), no. 2, 231-264. · Zbl 0654.12008
[22] M. Ram Murty and V. Kumar Murty, Mean values of derivatives of modular \(L\)-series , Ann. of Math. (2) 133 (1991), no. 3, 447-475. JSTOR: · Zbl 0745.11032 · doi:10.2307/2944316
[23] Jin Nakagawa and Kuniaki Horie, Elliptic curves with no rational points , Proc. Amer. Math. Soc. 104 (1988), no. 1, 20-24. JSTOR: · Zbl 0663.14023 · doi:10.2307/2047452
[24] Jan Nekovář, Class numbers of quadratic fields and Shimura’s correspondence , Math. Ann. 287 (1990), no. 4, 577-594. · Zbl 0679.12009 · doi:10.1007/BF01446915
[25] Ken Ono and Christopher Skinner, Fourier coefficients of half-integral weight modular forms modulo \(l\) , Ann. of Math. (2) 147 (1998), no. 2, 453-470. JSTOR: · Zbl 0907.11017 · doi:10.2307/121015
[26] K. Ono and C. Skinner, Nonvanishing of quadratic twists of modular \(L\)-functions , · Zbl 0937.11017
[27] Karl Rubin and Andrew Wiles, Mordell-Weil groups of elliptic curves over cyclotomic fields , Number theory related to Fermat’s last theorem (Cambridge, Mass., 1981), Progr. Math., vol. 26, Birkhäuser Boston, Mass., 1982, pp. 237-254. · Zbl 0519.14017
[28] Goro Shimura, The special values of the zeta functions associated with cusp forms , Comm. Pure Appl. Math. 29 (1976), no. 6, 783-804. · Zbl 0348.10015 · doi:10.1002/cpa.3160290618
[29] Glenn Stevens, Arithmetic on modular curves , Progress in Mathematics, vol. 20, Birkhäuser Boston Inc., Boston, MA, 1982. · Zbl 0529.10028
[30] Glenn Stevens, The cuspidal group and special values of \(L\)-functions , Trans. Amer. Math. Soc. 291 (1985), no. 2, 519-550. · Zbl 0579.10011 · doi:10.2307/2000098
[31] Glenn Stevens, The Eisenstein measure and real quadratic fields , Théorie des nombres (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 887-927. · Zbl 0684.10028
[32] Lawrence C. Washington, The non-\(p\)-part of the class number in a cyclotomic \({\mathbf Z}_{p}\)-extension , Invent. Math. 49 (1978), no. 1, 87-97. · Zbl 0403.12007 · doi:10.1007/BF01399512
[33] A. Wiles, On ordinary \(\lambda\)-adic representations associated to modular forms , Invent. Math. 94 (1988), no. 3, 529-573. · Zbl 0664.10013 · doi:10.1007/BF01394275
[34] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem , Ann. of Math. (2) 141 (1995), no. 3, 443-551. JSTOR: · Zbl 0823.11029 · doi:10.2307/2118559
[35] S. Wong, Elliptic curves of rank zero ,
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.