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Pacific Journal of Mathematics Vol. 281 (2016) Pacific Journal of Mathematics Volume 281 No. 2 April 2016 PACIFIC JOURNAL OF MATHEMATICS msp.org/pjm Founded in 1951 by E. F. Beckenbach (1906–1982) and F. Wolf (1904–1989) EDITORS Don Blasius (Managing Editor) Department of Mathematics University of California Los Angeles, CA 90095-1555 [email protected] Paul Balmer Vyjayanthi Chari Daryl Cooper Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California University of California Los Angeles, CA 90095-1555 Riverside, CA 92521-0135 Santa Barbara, CA 93106-3080 [email protected] [email protected] [email protected] Robert Finn Kefeng Liu Jiang-Hua Lu Department of Mathematics Department of Mathematics Department of Mathematics Stanford University University of California The University of Hong Kong Stanford, CA 94305-2125 Los Angeles, CA 90095-1555 Pokfulam Rd., Hong Kong fi[email protected] [email protected] [email protected] Sorin Popa Jie Qing Paul Yang Department of Mathematics Department of Mathematics Department of Mathematics University of California University of California Princeton University Los Angeles, CA 90095-1555 Santa Cruz, CA 95064 Princeton NJ 08544-1000 [email protected] [email protected] [email protected] PRODUCTION Silvio Levy, Scientific Editor, [email protected] SUPPORTING INSTITUTIONS ACADEMIA SINICA, TAIPEI STANFORD UNIVERSITY UNIV. OF CALIF., SANTA CRUZ CALIFORNIA INST. OF TECHNOLOGY UNIV. OF BRITISH COLUMBIA UNIV. OF MONTANA INST. DE MATEMÁTICA PURA E APLICADA UNIV. OF CALIFORNIA, BERKELEY UNIV. OF OREGON KEIO UNIVERSITY UNIV. OF CALIFORNIA, DAVIS UNIV. OF SOUTHERN CALIFORNIA MATH. SCIENCES RESEARCH INSTITUTE UNIV. OF CALIFORNIA, LOS ANGELES UNIV. OF UTAH NEW MEXICO STATE UNIV. UNIV. OF CALIFORNIA, RIVERSIDE UNIV. OF WASHINGTON OREGON STATE UNIV. UNIV. OF CALIFORNIA, SAN DIEGO WASHINGTON STATE UNIVERSITY UNIV. OF CALIF., SANTA BARBARA These supporting institutions contribute to the cost of publication of this Journal, but they are not owners or publishers and have no responsibility for its contents or policies. See inside back cover or msp.org/pjm for submission instructions. The subscription price for 2016 is US $440/year for the electronic version, and $600/year for print and electronic. Subscriptions, requests for back issues and changes of subscribers address should be sent to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163, U.S.A. The Pacific Journal of Mathematics is indexed by Mathematical Reviews, Zentralblatt MATH, PASCAL CNRS Index, Referativnyi Zhurnal, Current Mathematical Publications and Web of Knowledge (Science Citation Index). The Pacific Journal of Mathematics (ISSN 0030-8730) at the University of California, c/o Department of Mathematics, 798 Evans Hall #3840, Berkeley, CA 94720-3840, is published twelve times a year. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. POSTMASTER: send address changes to Pacific Journal of Mathematics, P.O. Box 4163, Berkeley, CA 94704-0163. PJM peer review and production are managed by EditFLOW® from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2016 Mathematical Sciences Publishers PACIFIC JOURNAL OF MATHEMATICS Vol. 281, No. 2, 2016 dx.doi.org/10.2140/pjm.2016.281.257 THE EISENSTEIN ELEMENTS OF MODULAR SYMBOLS FOR LEVEL PRODUCT OF TWO DISTINCT ODD PRIMES DEBARGHA BANERJEE AND SRILAKSHMI KRISHNAMOORTHY We explicitly write down the Eisenstein elements inside the space of modu- lar symbols for Eisenstein series with integer coefficients for the congruence subgroups 00. pq/ with p and q distinct odd primes, giving an answer to a question of Merel in these cases. We also compute the winding elements explicitly for these congruence subgroups. Our results are explicit versions of the Manin–Drinfeld theorem. 1. Introduction In his landmark paper on Eisenstein ideals, Mazur studied torsion points of elliptic curves over Q and gave a list of possible torsion subgroups of elliptic curves (see [Mazur 1977, Theorem 8]). Merel[1996b] wrote down modular symbols for the congruence subgroups 00.p/ for any odd prime p that correspond to differential forms of the third kind on the modular curves. He then used these modular symbols to give a uniform upper bound on the torsion points of elliptic curves over any number field in terms of its extension degree[Merel 1996a]. The explicit expressions of winding elements for prime level of[Merel 1996b] were used by Calegari and Emerton[2005] to study the ramifications of Hecke algebras at the Eisenstein primes. Several authors afterwards studied the torsion points of elliptic curves over number fields using modular symbols. In the present paper, we study elements of relative homology groups of the modular curve X0.pq/ that correspond to differential forms of the third kind with p and q distinct odd primes. As a consequence, we give an “effective” proof of the Manin–Drinfeld theorem (Theorem 9) for the special case of the image in H1.X0.pq/; R/ of the path in H1.X0.pq/; @.X0.pq//; Z/ joining 0 and i1. Since the algebraic parts of the special values of the L-function are obtained by integrating differential forms on these modular symbols, our explicit expression of the winding elements should be useful for understanding the algebraic parts of the special values at 1 of the L-functions of the quotient Jacobian of modular curves for the congruence subgroup 00.pq/ [Agashe 2000]. MSC2010: primary 11F67; secondary 11F11, 11F20, 11F30. Keywords: Eisenstein series, modular symbols, special values of L-functions. 257 258 DEBARGHA BANERJEE AND SRILAKSHMI KRISHNAMOORTHY For N 2 fp; q; pqg, consider the basis EN of E2.00.pq// (Section 4) for which all the Fourier coefficients at i1 belong to Z. The meromorphic differential forms EN .z/ dz are of the third kind on the Riemann surface X0.pq/ but of the first kind on the noncompact Riemann surface Y0.pq/. Let ξ V SL2.Z/ ! H1.X0.pq/; cusps; Z/ be the Manin map (Section 3). For any two coprime integers u and v with v ≥ 1, let S.u; v/ 2 Z be the Dedekind sum (see Section 4.1). If g 2 P1.Z=pqZ/ is not of the form .±1; 1/, .±1 ± kx; 1/ or .1; ±1 ± kx/ with x one of the primes p or q, then we can write it as .r − 1; r C 1/. Let δr be 1 or 0 depending on whether r is odd or even. For any integer k, let 0 0 sk D k C (δk − 1/pq be an odd integer. Choose integers s; s and l; l such that 0 0 l.sk x C 2/ − 2spq D 1 and l sk x − 2s pq=x D 1. Let 1 C 4spq −2l 1 C 4s0 pq=x −2l0 x;k D x;k D γ1 and γ2 0 0 −4s.sk x C 2/pq 1 C 4spq −4s .sk/pq 1 C 4s pq=x be two matrices (see Lemma 28). For l D 1; 2, consider the integers x;k x;k x;k x;k x;k x;k PN (γl / D sgn.t(γl // 2 S s(γl /; jt(γl /jN − S s(γl /; jt(γl /j − x;k 1 j x;k j C x;k 1 j x;k j S s(γl /; 2 t(γl / N S s(γl /; 2 t(γl / with x;k D − C x;k D − − C s(γ1 / 1 4spq.1 sk x/; t(γ1 / 2.l 2s.sk x 2/pq/ and 0 1 0 0 s(γ x;k/ D 1 − 4s pq s − ; t(γ x;k/ D −2.l − 2s s pq/: 2 k x 2 k 1 Define the function FN V P .Z=pqZ/ ! Z by 82.S.r; N/ − 2S.r; 2N// if g D .r − 1; r C 1/, > <>P (γ x;k/ − P (γ x;k/ if g D .1 C kx; 1/ or g D .−1 − kx; 1/, F .g/ D N 1 N 2 N − x;k C x;k D C D − − > PN (γ1 / PN (γ2 / if g .1; 1 kx/ or g .1; 1 kx/, > :0 if g D .±1; 1/. Theorem 1. The modular symbol D X EEN FEN .g)ξ.g/ g2P1.Z=pqZ/ in H1.X0.pq/; @.X0.pq//; Z/ is the Eisenstein element (Section 5) corresponding to the Eisenstein series EN 2 E2.00.pq//. In[Banerjee 2014], a description is given of Eisenstein elements in terms of certain integrals for M D p2. In this article, we give an explicit description in THEEISENSTEINELEMENTSOFMODULARSYMBOLS 259 x;k x;k V ! terms of two matrices γ1 and γ2 . Let B1 R R be the periodic first Bernoulli polynomial. For the Eisenstein series E pq (Section 4), we write down the Eisenstein elements more explicitly if g D .r − 1; r C 1/. Replacing p with pq [Merel 1996b, Lemma 4], we write pq−1 X hr F .r − 1; r C 1/ D B : pq 1 2pq hD0 Recall the concept of the winding elements (Definition 37). We write down the explicit expression of the winding elements for the congruence subgroup 00.pq/. Corollary 2. X n 1 o .1 − pq/e D F .1; x/ 0; : pq pq x x2.Z=pqZ/∗ Note that if ν D gcd.pq −1; 12/ and n D .pq −1)/ν, then a multiple of winding element nepq belongs to H1.X0.pq/; Z/. Manin and Drinfeld proved that the modular symbol f0; 1g belongs to H1.X0.N/; Q/ using the theory of suitable Hecke operators acting on the modular curve X0.N/=Q. In this paper, we follow the approach of Merel[1996b, Proposition 11].
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