2021 Fiscal Year Research-status Report
Newform theory for the full space via local Shimura correspondence and Waldspurger-type theorem
| Project/Area Number |
18K13396
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| Research Institution | Tokyo Institute of Technology |
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Principal Investigator |
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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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Keywords | Hecke algebra / Whittaker functions |
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| Outline of Annual Research Achievements |
We describe the Hecke algebra of G=GL_2(Q_p) modulo K_0=K_0(p^2) for any prime p. In an earlier work we described the case when p=2 and used this to give a complete description of the Whittaker function of new vectors, without using any realization of representations. In the current work we extend it to all odd primes. The complete set of representatives of G modulo K_0 modulo central elements is given by d(p^n) w(p^n)} for n integers, d(p^n)y(p) for n>=1, y(p)d(p^{-n}) for n >=1, y(p)w(p^n), w(p^n)y(p) and D_{n,a}= y(a p)w(p^n)y(p) for n>=2 where a runs between 1, p-1; and so corresponding characteristic functions X_g give a vector space basis of the Hecke algebra. We give a presentation of the Hecke algebra: the algebra is generated by U_1=X_{w(p)}, U_2=X_{w(p^2)}, D_{2,a}= X_{y(a p)w(p^2)y(p)}, D_{3,b}= X_{y(b p)w(p^2)y(p)} with a between 2 and (p-1)/2 and and b between 1 and (p-1)/2 with several defining relations, including (p-1) inductive ones of the type D 4,4a + 2sum_{d with certain condition}D 4,d = D 3,a D 3,1 + combinations of known terms, from which D_{4,a}’s are uniquely determined. Using some of these relations we also completely describe Whittaker new vectors of level p^2, again we do not use any realization of representation (principle series, Steinberg or supercuspidal) to obtain the Whittaker function. This is a joint work in progress with Moshe Baruch.
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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
Our approach towards local Shimura correspondence has been to describe Hecke algebra of PGL2 and double cover of SL_2 of certain levels using generators and relations and prove isomorphism between them. For level p^2, we have now the full PGL_2 Hecke algebra. We now need to do similar computations for double cover of SL_2. Regarding levels coming from powers of 2,
we have already computed double coset representatives (for both the cases), we will use them to get generators and relations.
Another motivation of our work has been to obtain information purely from the presentations of algebra. Our description of Whittaker new vectors, without using any realization of representations, is one such application.
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| Strategy for Future Research Activity |
The next plan of the project is three-fold: obtain the corresponding presentation for Hecke algebra of double cover of SL_2(Q_p) modulo K_0=K_0(p^2); in the case p=2, describe the Hecke algebra of G=GL_2(Q_p) modulo K_0=K_0(2^n) for any n and then obtain corresponding presentation for double cover Hecke algebra. We expect this to give us a local Shimura correspondence
that fits with Ueda’s theory of newforms of half-integral weight. In each of these cases we also plan to use the Hecke algebra relations to describe Whittaker function of the new vectors and then do local computations, as in Baruch-Mao, to obtain Kohnen-Zagier type correspondence.
There has been recent works by Sirolli-Tornaria on Waldspurger-Kohnen-Zagier type formula (extending Baruch-Mao’s work) for general levels except perfect squares, their construction of corresponding half-integral weight forms (which belongs to Kohnen plus space) is using certain linear combination of ternary theta series coming from quaternionic modular forms. As our earlier work suggests, these half-integral weight forms are not new in the whole space. Our approach has been to get such formula in terms of newforms of half-integral weight, the first step towards that direction will be to obtain Waldpurger-Baruch-Mao’s formula for our minus spaces (which are new in the whole space).
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| Causes of Carryover |
I plan to use a significant portion of the remaining Kakenhi for an international conference that we are organizing in March 2023. The conference will be on current advances in different areas of number theory.
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