On the critical values of Hecke L
Transcription
On the critical values of Hecke L
M ÉMOIRES DE LA S. M. F. B ENEDICT H. G ROSS D ON Z AGIER On the critical values of Hecke L-series Mémoires de la S. M. F. 2e série, tome 2 (1980), p. 49-54. <http://www.numdam.org/item?id=MSMF_1980_2_2__49_0> © Mémoires de la S. M. F., 1980, tous droits réservés. L’accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ SOCIETE MATHEMATIQUE DE FRANCE 2e serie, Mfimoire n0 2, 1980, p. 49-54 ON THE CRITICAL VALUES OF HECKE L-SERIES by Benedict H . GROSS and Don ZAGIER Let E be the elliptic curve over Q with minimal model y 2 + xy = x3 - x 2 - 2x - 1 . The modular invariant, discriminant, and conductor of E are given by 3 - -33 . 53 N = (72) Let on Q E . dx denote the fundamental real period of the Neron differential u) = ——— : 0 = J (D = 1.93331170561681... E(IR) Over the field <? = 7L\.—^""•~3 • Hence 0 Chowla and Selberg [l] : K = flK-^/^) , E has complex multiplication by can be determined explicitly, using an identity of 49 B . H . GROSS and D. ZAGJ^J? ^ ^ r(i/7)r(2/7)r(4/7) ^ . 2ni Similarly, the L- series of of K . The conductor of is prime to 7 E X is equal to the L-series of a Hecke character is t"® ideal (-/^) ; for Q^ an ideal of K X which : X(GO = a where 0 = (a) , a3 = Krnod ^p?) We have calculated the central critical values of the Hecke L-series which are associated to odd powers of the character X • Let n ^ 1 be an integer ; the Dirichlet series Ux^s)^^2 ^ s IN 0 converges absolutely in the right half-plane Re(s) > n •»• ^ . It extends -to a holomorphic function on the entire complex plane : the modified function A^2""1^) = (7/27c)s F(s) UX211''1^) satisfies Hecke's functional equation ACX 2 "' 1 ^) = (-I)"'1 A(X2n-l,2n-s) It follows that the value of strip, vanishes when When n n Ux2""1,®) at s = n : . , the center of the critical is even. is odd, define a n by _ ^2n-l , Q211"1 ^__ L(X n) « ————————r . .. , . n l (n l)s (2^/^) • (We found this normalization by trial and error , it is consistent with the work of Katz on the interpolation of real analytic Eisenstein series [.2].) The values of a for 1 ^ n ^ 33 are listed in Table 1 . n 50 HECKE L-SERIES Table 1 n a n 1 1/2 3 2 5 2 7 2(3) 2 9 2(7) 2 11 2(3 2 .5 13 2(3.7.2 15 2(3.7.3 17 2(3.5.' 19 2(3 3 .7 21 2(3 2 .5 23 2(3 3 .5 25 2(3 2 .7 S 27 2(3 2 .5 , 2(3 .5 29 2(3 5 .5 , 2(3 .5 31 33 Let p s Kmod 4) be a prime and let \ be the Hecke character P X (Q) = (wa) x(Q) P P The Hecke L-series L(x »s) which becomes isomorphic to . is equal to the L-series of an elliptic curve E over <(->/?) 51 . Let 0 = 0/^/p P and define E /<0 a n by B . H . GROSS and D. ZAGJ^R ^2n-l "p - / 2n-l . L(x n) ' • (W^)"-1 ^ • - Again, a p n vanishes when (p) n n is even. For we found the values listed in Table 2. 52 n odd and p « 5> 13> 17> 29, 53 CM C^ CM ?' •n I <—' Si «^ in u ^t • «J« 0 CM ^ c^ • (-» 't^ «' r^ CM w\ <^ -a* CM w\ Q <^ ^M 'CM 'CM CM CM CM t^ CM * U"\ ^ CM <»\ 01 ? ^-s CM \0 S' 0>* CM ? 0 CM CM CM c"t CM CM (^ CM CM CM ^M CM P Q CM CM CM CM ^ CM^ t^ «^ CM CM ^ »-i (^ (^ ^ »-• vo CM CM CM ^« t^ S »-• a> 'sO \D • CM CM^ c^ • ir\ CM • [S. (<^ CM CM CM CM CM ^-s r" CM • -3" (^ ^ »-t co CM 0 ^ CM ^ (»«. «^ • r^ f^ • »-< t^ CM CM CM CM CM CM CM CM CM CM CM CM CM -^ CM CM CM r" ^ v^ CM ^ IT\ CM C O-y^l C M C C M m M C C M r^ ^ t^ . t^ i1^ • M o» ^ . IT\ C C M o^ 53 C C M ^ »-< C C M M <^ »-i t^ • CM V\ (^ M t> v£> CM r• CM V\ tr\ M <^ • 0^ CO r• CM • ir\ M ^ \£> C^ (^ -3V\ p^ C C M M m »-• (^ C C M M t^ ^ B.H. Note : GROSS and D. ZAGJ£J? In all the cases we computed, twice a square, depending on whether (?) is ap -1 or is either a square or +1 . Can one prove this is general ? Are there "higher Tate-Shafarevitch groups" associated to the abelian varieties (E ) " whose orders can be conjecturally related to P these groups carry a natural alternating pairing ? Bibliography [l] a ? : CHOWLA S . , and SELBERG A . , On Epstein*s Zeta Function. J . Crelle 227 ( 1 9 6 7 ) , 96-110. [2] Do " KATZ N . , p-adic interpolation of real analytic Eisenstein series. Annals Math. 104 ( 1 9 7 6 ) , 459-571. B . Gross Princeton University Fine Hall Princeton, N . J . 08540 ( U . S . A . ) D . Zagier Mat. Inst. d . Univ. Bonn Wegelerstrasse 10 53 Bonn ( B . R . D . ) 54