2015 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 / Half integral weight / Metaplectic group |
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| Outline of Annual Research Achievements |
Let Q_4=W_4U_4 be the Niwa operator on S_{k/2}(4), it has two eigenvalues,e_1,e_2. Kohnen plus space S_4 is the e_1 eigenspace of Q_4 and is Hecke isomorphic to S_{2k}(1). We consider Q'_4=U_4W_4. Let A_4 be the e_1 eigenspace of Q'_4. We show A_4 is also Hecke iso. to S_{2k}(1); this gives iso. between direct sum of S_4 and A_4 and that of S_{2k}(1) and V(2)(S_{2k}(1)). Let S'_4 be the intersection of e_2 eigenspaces of Q_4 and Q'_4. We show S'_4 is Hecke iso. to S_{2k}{new}(2). We call S'_(4) the minus space at level 4. Recall earlier we proved S_{2k}{new}(2) is intersection of -1 eigenspaces of two operators. We want to define minus space for a general level 4N and expect it to be iso. to S_{2k}{new}(2N). So we study the operators on S_{k/2(4N) that comes from local Hecke algebra of the metaplectic group. For a prime p, Let p_n denotes nth power of p. Let G_p be a double cover of SL_2(Q_p) defined by a certain 2-cocycle. Let H(G_p//K_0(p_n),γ) be the Hecke algebra of G_p corres. to K_0(p_n) (inv. image of Γ_0(p_n)Z_p under covering map) and a genuine central character γ. We separately study the case of p odd and p=2 as double cover splits over SL_2(Z_p) for p odd but does not when p=2. Loke and Savin studied H(G_2//K_0(4),γ) (γ has order 4) and use it to obtain Q_4. We extend their work to H(G_2//K_0(8), γ) (with two choices ofγ, each order 4) and H(G_p//K_0(p),γ) (with γ quadratic) for odd prime p and describe them by generators and relations. We translate these local operators to classical operators and obtain U(p_2) as well as Ueda's involution in odd p case.
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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
We wanted to generalize Loke and Savin's work of description of H(G_2//K_0(4),γ). While understanding their work we realized that in order to work with a general prime power p_n, we must develop a general theory for computing support of our local Hecke algebra. We gave a criteria for support of H(G_p//K_0(p_n),γ) in terms of value of the central character γ on certain commutators. We further simplified checking of this criteria by using triangular decomposition of K_0(p_n) and it's certain subgroup. While doing this we had to take extra care of the cocycle multiplication. We could use this theory to obtain the generators of H(G_2//K_0(8),γ) and H(G_p//K_0(p),γ) and compute relations amongst them.
In an another joint work with Dr. J. C. Lario we studied π/6 congruent numbers. We gave two definitions for a natural number n to be π/6 congruent, one relating to a rational elliptic curve while other relates to an elliptic curve defined over quadratic field Q(\sqrt{3}). Denote the Bth power of A by A_B (A can be a variable and B is a natural number). Then using the first definition we obtain that n is π/6 congruent iff Mordell-Weil rank of elliptic curve Cn: Y_2 = X_3 + 6nX_2 - 3n_2X over rationals is >=1. In fact C1 turns out to be isogenous to j-invariant 0 curve Y_2 = X_3 +1 and using Waldspurger's theorem we could give a Tunnell like criterion for n to be π/6 congruent for n in certain congruence classes. We realized that some of our work on j-invariant 0 curve coincides with that of G. Frey's work but some of our conjectures are still open. This is work in progress.
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| Strategy for Future Research Activity |
We plan to describe local Hecke algebra H(G_p//K_0(p_n), γ) for a general prime p and positive integers n. We shall first restrict to the subalgebra supported on the inverse image of SL_2(Z_p). In order to describe full Hecke algebra we would need to deal with the double coset representatives of G_p modulo K_0(p_n) the number of which increases with n making computations more difficult. Also when p is odd we would like to restrict ourselves to quadratic central character. This is because when γ is not quadratic then H(G_p//K_0(p), γ) might not be supported on certain double coset representatives that gives us involution when translated to their classical counterparts. Once we describe these algebras we will translate the local operators into classical operators on the space S_{k/2}(4N, χ). We shall study eigenspaces of these operators and define minus subspace (as we did in the level 4 case) as an intersection of certain eigenspaces. We expect our minus space at level 4N to be Hecke isomorphic to the new subspace of S_{2k}(2N). In the process we also expect to obtain Ueda's newform theory for half integral weight space. Note that Ueda's work does not apply to general level 4N. In particular we expect to obtain Ueda's twisting operators once we study H(G_p//K_0(p_n)) for n > 1. Once we have described the minus space we would like to apply to the forms in minus space the adelic Waldspurger formula given by Baruch and Mao. We expect 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.
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