| 研究課題/領域番号 |
18K13396
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| 研究機関 | 東京工業大学 |
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研究代表者 |
プルカイト ソーマ 東京工業大学, 理学院, 特任准教授 (30806592)
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| 研究期間 (年度) |
2018-04-01 – 2022-03-31
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| キーワード | Local Hecke algebra / Automorphic forms / Half-integral weight / Newforms / Waldspurger's formula |
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| 研究実績の概要 |
We want to describe local Hecke algebras of the non-trivial central extension of SL_2(Q_p) by ±1, corresponding to the (inverse image of) open compact subgroups K_0(p^n) and certain genuine characters of K_0(p^n). In the first two papers listed below we described the cases: p odd, n=1 and, p=2, n=3. For p=2, general n we have obtained a complete (infinite) set of double coset representatives for SL_2(Q_p) modulo K_0(2^n) in terms of diagonal, antidiagonal and unipotent lower triangular matrices. We consider two categories while working with the coset representatives, the first category which consists of infinitely many representatives is determined completely by the double coset representatives (finite) of the maximal compact modulo K_0(2^n), the remaining case consists of finitely many representative and we have a computable algorithm for it. Our methods can be generalized to any prime p.
For the Kohnen-Zagier formula for the minus space of level 4M, we computed the Weil 2-adic constant, the delta function and the inner products. We used local (PGL_2) Hecke algebra relations to impose specific conditions on Kirillov-Whittaker functions of new vectors of level 4. These are essential for computing local factors in Baruch and Mao’s adelic formula.
Loke and Savin use action of the “generating” central element of the level 4 Hecke algebra to determine the Steinberg representation. We described the center of level 8 Hecke algebra. We found that unlike the level 4 case, we have two “generators”: one is of infinite order like in Loke-Savin while the other is an idempotent.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The progress that we have made this year is as expected. We were able to complete the first step towards obtaining presentation of local Hecke algebra of level 2^n which would also generalize for level p^n. To compute the support we are concentrating on quadratic characters and there are two of them as in the case of level 8 Heck algebra. We have obtained some partial results in this direction.
While computing the local factors in the Baruch and Mao formula we made some new observations that were not there in the earlier papers of Baruch and Mao, and that would make our computation of local factors easier.
The interesting observations that we made on imposing conditions on representations where we can find the new vectors corresponding to the minus space would be significant for obtaining general Waldspurger-Kohnen-Zagier type formula.
Based on our earlier paper on integral weight forms, we also relooked at how local Hecke algebra relations could be used to distinguishing representations of different conductors.
A considerable time was spent in improving our earlier article on minus space on level 4M and 8M, on the basis of referee’s questions and our observations, we added a few more results and remarks in papers on level 4M and 8M and now the current version is accepted and is in the publication queue.
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| 今後の研究の推進方策 |
We will give presentation of the genuine local Hecke algebras of the double cover of SL_2(Q_2) corresponding to K_0(2^n) and quadratic characters of K_0(2^n). We will use our machinery to check the support of the double coset representatives that we computed. As noted earlier we divide the coset representatives into two categories and we expect that there won’t be any support on the second category. We have done some basic computations for relations, we will complete that work which should give us the defining generators and relations. Next step is to do similar computation for local Hecke algebras of PGL_2 with trivial character and compare both. We expect local Shimura correspondences in line with Ueda’s (global) work. We will use local Hecke algebra relations so obtained to impose specific conditions on Kirillov-Whittaker functions of new vectors which should make it amenable to compute the local factors in Baruch and Mao’s formula. Our work shows that the local representation where a Whittaker function of level 4 belongs can be either principle series, Steingberg or supercuspidal, which gives another proof of what is already known, but it also imposes conditions on the representations, for example, the types of supercuspidals etc. Interestingly, our computation only uses the relations of the subalgebra. Since we have full description of the level p^n subalgebra, we next plan to obtain similar conditions for level p^n.
Following Loke-Savin, we will also explore the role of the “new generator” of the center of level 8 Hecke algebra on the local representation theory.
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| 次年度使用額が生じた理由 |
The usage of budget is lesser than expected since due to teaching obligations, I could only do limited travelling and some of these travels were funded by the host of the institutes where I visited. For the next year, I will be inviting my collaborator to Japan as well and may use my budget to support his stay here.
I will do a business trip to Oberwolfach in December and then may visit Utah in March for automorphic forms workshop. Apart from these travels, I am planning to organize a number theory workshop either during Spring or summer 2020 and I intend to use my budget for the same.
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