CHUNG PANG MOK. The correct form of Proposition 2.2 of [M1] should be stated as follows:

Transcription

1 CORRIGENDUM AND ADDENDUM TO SPECIAL VALUES OF L-FUNCTIONS OF ELLIPTIC CURVES OVER Q AND THEIR BASE CHANGE TO REAL QUADRATIC FIELDS [J. NUMBER THEORY 130 (2010), NO. 2, ] CHUNG PANG MOK Abstract. In this note we make corrections to the above mentioned paper. In this note, we continue to use the notations in [M1] and refer to loc. cit. for the setup. Here we only remind that E{Q is an elliptic curve whose conductor is equal to Mp, where p is a prime that does not divide M. We also consider a real quadratic field satisfying the condition that p is inert in F, and that all primes dividing M split in F. Denote by ψ the even quadratic Dirichlet character that corresponds to the real quadratic field F. The correct form of Proposition 2.2 of [M1] should be stated as follows: Proposition 1. (Correction of Proposition 2.2 of [M1]). Assume the conductor c δ of δ is relatively prime to 2. If the rank 0 case of the Birch and Swinnerton-Dyer conjecture is true for r E{F, then the value L alg p1, E{F, δq is the square of a rational number, up to a factor of two. We refer to page 432 of [M1] for the conditions that the quadratic Hecke character δ of the real quadratic field F has to satisfy. There are several points that have to be corrected in the proof of loc. cit. 1. For the prime p po F, the curve r E{F is non-split multiplicative at p. In this case the local Tamagawa factor c p p r E{F q is equal to 1 or For the primes l dividing c δ, the curve r E{F has additive reduction of type I 0. In this case the local Tamagawa factor c l p r E{F q is equal 1, 2 or Mathematics Subject Classification. 11G40,11G05. 1

2 2 CHUNG PANG MOK For discussions related to computation of local Tamagawa factors of elliptic curves under quadratic twists, the reader may refer to [R] for example. 3. The period factor Ω re{f : 1 pn F {Q c δ q 1{2 Ω E{F 1 pn F {Q c δ q 1{2 pω E{Qq 2 as defined on page 432 of [M1], is up to a factor of 2, equal to the period factor for the statement of the Birch Swinnerton-Dyer conjecture for re{f. This factor of 2 is overlooked in the discussions in [M1]. With these corrections being taken into account, the proof of Proposition 2.2 of [M1], stated in the corrected form as Proposition 1 above, is then valid. As for Proposition 2.4 of [M1], the correct reference for the proof should be Corollary 6.7 of [M2]: Proposition 2. (Proposition 2.4 of [M1]; for proof see Corollary 6.7 of [M2]). Assume that E{Q is split-multiplicative at p, and that E has multiplicative reduction at some prime other than p. Suppose further that: L 1 p1, E{Qq 0, Lp1, E{Q, ψq 0. Then for δ considered in the context of Proposition 2.2 of [M1], the quantity 1 2 Lalg p1, E{F, δq is the square of a rational number. Another clarification is that the sign of the functional equation for Lps, E{F q is always equal to 1 (because Lps, E{F q Lps, E{Qq Lps, E{Q, ψq and the signs of the functional equations for Lps, E{Qq and Lps, E{Q, ψq are opposite to each other). The sign of the functional equation for Lps, E{F, δq, for δ considered in Proposition 1 above, is opposite to that of Lps, E{F q, hence is equal to `1. In section 3 of [M1] we made the hypothesis that L 1 p1, E{Qq 0 and Lp1, E{Q, ψq 0. This hypothesis is equivalent to the condition that both Lps, E{Qq and Lps, E{F q vanish to order 1 at s 1. This hypothesis is needed for the computations of section 3, but for the statement of Conjecture 3.1 of [M1], this is not necessary. We will make the correct version of Conjecture 3.1 of loc. cit. in Conjecture 3 below. Finally concerning the computations on pages of [M1]:

3 CORRIGENDUM/ADDENDUM 3 1. Equations (3.15) and (3.16) of [M1], concerning the formula of Gross- Zagier [GZ] and its generalization as given by Zhang [Z1,Z2], when being applied to E{L and E{K respectively, should be corrected as follows. Firstly for (3.15) of loc. cit. the correct formula should be L 1 p1, E{Lq 1 D 1{2 L xf E, f E y deg E{Q htpp q where deg E{Q is the degree of a (fixed) parametrization of E{Q by the modular curve X 0 pmpq{q. The factor deg E{Q is missing in equation (3.15) of [M1]. 2. Similarly equation (3.16) of loc. cit. should be corrected as: L 1 p1, E{Kq 1 xf E, f E y htppq pn F {Q D K{F q 1{2 deg E{F where deg E{F is the degree of a (fixed) parametrization of E{F by XpMO F, pq{f, the Shimura curve of Eichler level MO F associated to the quaternion algebra over F that ramifies at p and one of the two archimedean place of F (if we fix an embedding ι : F ãñ R, then the archimedean place of F where the quaternion algebra ramifies may be taken to be ι σ, where σ is the Galois conjugation of F over Q). The factor deg E{F is missing in equation (3.16) of [M1]. For more discussions related to the explicit form of generalizations of Gross-Zagier type formulas, the reader may refer to [CST]. 3. The L-value Lp1, E{ r K, δ rk q in equation (3.5) of [M1], is non-zero if and only if Lp1, E{F, δq is non-zero (indeed the factor Lp1, E{F, δ 1 q in equation (3.5) of loc. cit. is non-zero by the choice of δ 1 ; c.f. equation (3.4) of loc. cit.) Consequently equation (3.19) of loc. cit. should be corrected as follows: If Lp1, E{F, δq 0, then one has : Lp1, E{ K, r 1 xf E, f E y δ rk q pn F {Q D rk{f q 1{2 xφ E, Φ E y mod pqˆq 2 With these corrections, the end result of the computations on pages of [M1], which is given as equation (3.21) of loc. cit. should be corrected as follows:

4 4 CHUNG PANG MOK If Lp1, E{F, δq 0, then one has L alg p1, E{F, δq D1{2 F Lp1, E{F, δqpn F {Qc δ q 1{2 pω E{Q q 2 deg E{F xφ E, φ E y mod pqˆq 2. xφ E, Φ E y deg E{Q Recall that the sign of the functional equation for Lps, E{F, δq is equal to `1. Thus by the main result of Friedberg-Hoffstein [FH], there exists infinitely many δ such that Lp1, E{F, δq 0. Choose such a δ. Then since one expects, by Proposition 1 above, that L alg p1, E{F, δq is the square of a non-zero rational number up to a factor of two, we see that one also expects the right hand side of the above equation is also the square of a non-zero rational number, up to a factor of two. Consequently, equation (3.22) of [M1] should be stated as: up to a factor of 2. xφ E, Φ E y deg E{F? xφ E, φ E y deg E{Q mod pqˆq 2 We now state the correct form of Conjecture 3.1 of [M1] and in a slightly more general form where the conductor of E{Q is allowed to be of the form MQ, where the role of M is as before, while Q is a product of an odd number of distinct primes, such that all primes dividing Q are inert in F (the computations in section 3 of [M1] generalizes to this setting). Conjecture 3. (Corrected form of Conjecture 3.1 of [M1]). Let E{Q be an elliptic curve, whose conductor is of the form MQ, where pm, Qq 1 and Q is a product of an odd number of distinct primes. Denote by f E the normalized weight two eigenform of level MQ that corresponds to E{Q. Denote by deg E{Q the degree of a (fixed) parametrization of E{Q by the modular curve X 0 pmqq{q. Let F be a real quadratic extension of Q, such that all primes dividing M split in F, while all primes dividing Q are inert in F. Let f E be the base change of f E from Q to F ; thus f E is a parallel weight two Hilbert eigenform over F of level MQO F that corresponds to E{F. Denote by deg E{F the degree of a (fixed) parametrization of E{F by XpMO F, QO F q{f, the Shimura curve of Eichler level MO F associated

5 CORRIGENDUM/ADDENDUM 5 to the quaternion algebra over F that ramifies exactly at the primes of F dividing QO F and at one of the two archimedean places of F. Denote by B the definite quaternion algebra over Q, that ramifies at the primes dividing Q and the archimedean place. Let φ E be a scalarvalued automorphic eigenform with respect to the group Bˆ of Eichler level M, that corresponds to f E under the Jacquet-Langlands correspondence. Similarly let Φ E be a scalar-valued automorphic eigenform with respect to the group pb b Q F qˆ of Eichler level MQO F, that corresponds to f E under the Jacquet-Langlands correspondence (note that B b Q F ramifies exactly at the archimedean places of F and is split at all the finite places of F ). We normalize φ E and Φ E so that they take values in Q (with this condition φ E and Φ E are uniquely determined up to Qˆ-multiples). Denote by xφ E, φ E y the Petersson inner product of φ E with itself, and similarly for xφ E, Φ E y (these Petersson inners products are normalized as on p. 901 of [M2]). Then conjecturally one has: up to a factor of 2. xφ E, Φ E y deg E{F xφ E, φ E y deg E{Q mod pqˆq 2 In particular, by virtue of Proposition 2.4 of [M1] (stated as Proposition 2 above), one has that the conjecture is valid, at least under the following additional conditions on E{Q: The conductor of E{Q is of the form Mp, with all primes dividing M split in F and p is a prime that is inert in F (i.e. Q p), such that E{Q has split multiplicative reduction at p, and has multiplicative reduction at some prime dividing M. L 1 p1, E{Qq 0 and Lp1, E{Q, ψq 0 (where as before ψ is the quadratic Dirichlet character corresponding to F ). References [CST] L. Cai, J. Shu, Y. Tian, Explicit Gross-Zagier and Waldspurger formulae. Algebra and Number Theory 8 (2014), no. 10, [M1] C. P, Mok, Special values of L-functions of elliptic curves over Q and their base change to real quadratic fields. J. Number Theory 130 (2010), no. 2, [M2] C. P. Mok, Heegner points and p-adic L-functions for elliptic curves over certain totally real fields. Comment. Math. Helv. 86 (2011), no. 4,

6 6 CHUNG PANG MOK [FH] S. Friedberg, J. Hoffstein, Nonvanishing theorems for automorphic L- functions on GLp2q. Ann. of Math. (2) 142 (1995), no. 2, [GZ] B. Gross, D. Zagier, Heegner points and derivatives of L-series. Invent. Math. 84 (1986), no. 2, [R] K. Rubin, Fudge factors in the Birch and Swinnerton-Dyer conjecture. In Ranks of elliptic curves and random matrix theory, , London Math. Soc. Lecture Note Ser., 341, Cambridge Univ. Press, Cambridge, [Z1] S. W. Zhang, Gross-Zagier formula for GL 2. Asian J. Math. 5 (2001), no. 2, [Z2] S. W. Zhang, Gross-Zagier formula for GLp2q. II. In Heegner points and Rankin L-series, , Math. Sci. Res. Inst. Publ., 49, Cambridge Univ. Press, Cambridge, Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN address: mokc@purdue.edu