2014 Fiscal Year Annual Research Report
| Project/Area Number |
14F04319
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| Research Institution | Kyushu University |
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Principal Investigator |
金子 昌信 九州大学, 数理(科)学研究科(研究院), 教授 (70202017)
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| Co-Investigator(Kenkyū-buntansha) |
PURKAIT Soma 九州大学, 数理(科)学研究科(研究院), 外国人特別研究員
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| Project Period (FY) |
2014-04-25 – 2017-03-31
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| Keywords | Hecke algebras / Hecke operators / New forms / New vectors |
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| Outline of Annual Research Achievements |
We characterize the space of new forms of integer weight k and level M as a common eigenspace of certain operators with a given eigenvalue which depend on primes dividing M. Fix a prime p. Let p_k be the kth power of p. We study a certain Hecke algebra of functions on K=GL_2(Z_p) which are compactly supported, bi-invariant with respect to the subgroup K_0(p_n) that consists of elements with c-entry in p_nZ_p. We find generators and relations for this p-adic Hecke algebra and show it is commutative. We study the finite dimensional representations of K containing a K_0(p_n) fixed vector. Casselman showed that there is a unique irreducible representation of K which contains a K_0(p_n) fixed vector but no K_0(p_k) fixed vector for k<n. Such a vector is called a new vector. We explicitly describe these new vectors and action of our Hecke algebra on such vectors. We translate the operators in this algebra into classical operators and use these to classify the new space. Let M be a square-free positive number. For any prime p dividing M, let R_p = p_{k-1}U_pW_p and R_p' be conjugate of R_p by W_p where W_p is the Atkin-Lehner operator and U_p is the usual U operator. Then the new space at level M is precisely the intersection of the -1 eigenspaces of R_p and R_p' as p varies over the prime divisors of M. In the case when p_2 divides M, we again use a certain product of the W_{p_2} and U_p for a similar characterization. When p_3 divides M we introduce a family of operators that comes from the characteristic function of y(p_r) which capture the new and various spaces of old forms.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
In their remarkable work on the space of half integral weight modular forms Niwa and Kohnen considered a certain Hecke operator Q which is a composition of classical Hecke operators. Kohnen defined the plus space to be a particular eigenspace of this operator. Loke and Savin interpreted Kohnen's definition representation theoretically in the context of a Hecke algeba for the double cover of SL_2(Q_2) corresponding to the double cover of K_0(4) and used this Hecke algebra to classify the representations that contain maximal level vectors fixed by a certain congruence subgroup. In our research plan we wanted to extend their work by considering K_0(p_n) for a general prime p and a positive integer n. To gain a better understanding we tried to apply this method in the integral weight setting and we obtained characterization results for the new spaces of integral weight as well as various spaces of old forms. This work suggests that that there should be a similar theory of new forms for half integral weight where one can characterize the new space as a common eigenspace of certain operators that are obtained from the Hecke algebra of the double cover. Moreover we expect that the Shimura correspondence between the spaces of half integral and integral weight forms preserves the new spaces as characterized above.
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| Strategy for Future Research Activity |
We plan to study the Hecke algebra of genuine functions on double cover of SL_2(Q_p) that is compactly supported and quasi bi-invariant with respect to the inverse image of K_0(p_n) under the covering map and a central character g. We need to separately study the case of odd and even primes since the double cover splits over SL_2(Z_p) for p odd but does not split when p=2. First we shall consider Hecke algebras corresponding to K_0(2_n) for n >2. We have obtained a set of generators and relations for the K_0(8) case with two choices of central character. We plan to classify the representations of the maximal compact subgroup with a (K_0(8),g) eigenvector. We expect that these results will allow us to develop new form theory of half integral weight on Γ_0(8M) where M is odd and in general for level 2_nM. We intend to study similar Hecke algebras for odd primes and combine these results to give a new form theory for half integral weight forms for any level. In the process we expect to obtain new families of Hecke operators as we did in the case of integral weight. We plan to study these operators and compute trace formula for them. These should give us a Hecke isomorphism like the Shimura correspondence between the spaces of half integral and integral weight forms that preserves the new spaces on both sides. We plan to apply these results to obtain a Waldspurger type formula for the twisted L-values of an integer weight modular form of level 4M and 8M with M odd. The idea is to apply the adelic formula of Baruch and Mao in these cases in order to obtain an explicit formula.
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Research Products
(4 results)
