W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175--198.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....ranks 2 and higher should be infrequent. We will give a more precise conjecture about the frequency of twists with ranks at least 2. Sarnak has predicted that VE (x) should be about x 3=4 . His reasoning has to do with the formulas of Waldspurger [Wa] Shimura [Sh] and Kohnen and Zagier [KZ] which relate the value of LE (1=2; d ) to the Fourier coefficient of a half integral weight modular form. Roughly, 6) LE (1=2; d ) E c E (jdj) 2 = p d where E depends only on E and where the integers c E (jdj) are the Fourier coefficients of a half integral weight form. The Ramanujan ....

Kohnen, W. and Zagier, D., Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175--198.

....des couples suivants (s; t) dx; Gammady) dx; dz) dy; Gammadx) dy; dz) avec ici d 2 Z. La pr esence du facteur d dans les couples ci dessus correspond en fait a des tordues quadratiques. Soient s 0 et t 0 deux entiers sans facteurs carr es communs. Un r esultat du a Kohnen et Zagier [17] implique en effet l existence de tordues quadratiques E 1 (ds 0 ; dt 0 ) et E 2 (ds 0 ; dt 0 ) de rangs nuls, avec jdj N 2 1 , o u N 1 est le conducteur commun de E 1 (s 0 ; t 0 ) et E 2 (s 0 ; t 0 ) Si on admet l Hypoth ese de Riemann, alors d v erifie jdj N 1 (voir [14] ou [32] La ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175--198. COURBES DE FREY-HELLEGOUARCH 25

....Q , then L f j (k=2) 6= 0 for all j. Maeda has conjectured that the Hecke algebra of SL 2 (Z) is simple, which implies that the condition in Theorem 1 always holds. It appears to be possible that the characteristic polynomial is irreducible for all n. Theorem 1 is also stated by Kohnen and Zagier [KZ]. We have veri ed the irreducibility of the characteristic polynomial of T 2 acting on S k (1) for k 500, yielding Theorem 2. L f j (k=2) 6= 0 for k 500, k 0 mod 4. Questions concerning the characteristic polynomials of Hecke operators have received considerable attention lately. For ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175-198. J.B. CONREY D.W. FARMER

....p;k (x) the characteristic polynomial of the Hecke operator T p acting on S k (N; Let S k (N) S k (N; 1) and set T p;k = T 1;1 p;k . A conjecture of Maeda asserts that T p;k (x) is irreducible and has full Galois group over Q . This conjecture is related to the nonvanishing of L functions [KZ][CF] and to constructing base changes to totally real number elds for level 1 eigenforms [HM] Maeda s conjecture has been checked for all primes p 2000 and weights k 2000 [B] CF] FJ] The methods which have been used to check Maeda s conjecture involve computing the factorization of T 2;k ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175-198.

....algebra of S k (1) over Q is simple, and that its Galois closure over Q has Galois group the full symmetric group. There is even some speculation that T p;k (x) is irreducible in Q [x] and has full Galois group over Q for every prime p. This conjecture is related to the nonvanishing of L functions [KZ][CF] and to constructing base changes to totally real number elds for level 1 eigenforms [HM] There has been some progress towards this conjecture in recent years. For instance, Maeda s conjecture has been checked for p = 2 and k 540 [B] CF] and various other small cases. Also, we know the ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175-198.

....p#k (x) the characteristic polynomial of the Hecke operator T p acting on S k (N#) Let S k (N) S k (N#1) and set T p#k = T 1#1 p#k . A conjecture of Maeda asserts that T p#k (x) is irreducible and has full Galois group over Q . This conjecture is related to the nonvanishing of L functions [KZ][CF] and to constructing base changes to totally real number fields for level 1 eigenforms [HM] Maeda s conjecture has been checked for all primes p 2000 and weights k 2000 [B] CF] FJ] The methods which have been used to check Maeda s conjecture involve computing the factorization of T 2#k ....

W. Kohnen and D. Zagier,Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175--198.

....Primary 11F33, Secondary 11F67. 2 JAN H. BRUINIER v (a(d) 0. Note that we do not need the notion of a good modular form. Theorem 2 and Theorem 3 contain certain refinements. In the last section we will briefly indicate some applications. By the works of Waldspurger [Wa] Kohnen and Zagier [KoZa, Koh2] the results above have interesting consequences for the study of critical values of twisted L series attached to newforms of weight 2k (Theorem 5) Moreover, one can consider the Cohen Eisenstein series of level 4 and weight k 1=2 to find indivisibility results for special values L(1 Gamma k; ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 173-198.

....of all Hecke operators T (q 2 ) Let p 1 ; p r be distinct primes not dividing N and 1 ; r 2 f Sigma1g. Then there exist infinitely many square free integers d with a(d) 6= 0 and i d p j j = j for j = 1; r. By the work of Waldspurger [Wa] Kohnen and Zagier [KoZa, Koh2], Theorem 2 implies a non vanishing result for the central critical values of twisted L series attached to newforms of weight 2k. However, this also easily follows from the more general theorem of Friedberg and Hoffstein [FrHo] see also [BFH] MuMu] for related results) Acknowledgement. The ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 173-198.

....(x) the characteristic polynomial of the Hecke operator T p acting on S k (N; Let S k (N) S k (N; 1) and set T p;k = T 1;1 p;k . A conjecture of Maeda asserts that T p;k (x) is irreducible and has full Galois group over Q . This conjecture is related to the nonvanishing of L functions [KZ][CF] and to constructing base changes to totally real number fields for level 1 eigenforms [HM] Maeda s conjecture has been checked for all primes p 2000 and weights k 2000 [B] CF] FJ] The methods which have been used to check Maeda s conjecture involve computing the factorization of T 2;k ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175--198.

....# is unramified, this formula is due to Gross [37] and generalized by Hatcher [41] to modular form of high weights. In his thesis at Columbia, H. Xue [92] has obtained a full generalization to forms of high weight. Some related formula have been also obtained by Waldspurger [89] KohnenZagier [55], and Katok Sarnak [49] Nonvanishing of periods In view of the BSD conjecture and Mazur s conjecture one should have the Conjecture 8.8 (Generalized Mazur s conjecture) Let # be a finite subset of F . Then for all but finitely many ring class characters # unramified outside of #, one has ....

W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), 175--198.

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