| 研究実績の概要 |
Let G=GL_2(Q_p), K_0=K_0(p^n). In the Iwahori case (n=1), the description of the Hecke algebra is well-known. For general n, in an earlier work, we gave a description of the subalgebra supported on the maximal compact subgroup. We aim to describe local Hecke algebra of G modulo K_0(p^2). Let T_n=X_{d(p^n)}, U_n=X_{w(p^n)} for n integers, Z=X_{z(p)}, V_n=X_{d(p^n)y(p)} for n>=1, V=V_0=X_{y(p)}, A_n=X_{y(p)d(p^{-n})} for n >=1$, B_n=X_{y(p)w(p^n)}, C_n=X_{w(p^n)y(p)} and D_{n,a}= X_{y(a p)w(p^n)y(p)} for n>=2 where a runs between 1, p-1; X_g denotes the characteristic function of double coset of g. These form a vector space basis of the Hecke algebra, we compute relations amongst these to give the presentation. Using these relations, in the case p=3, we get that U_1, U_2, D_{3,1} generate the Hecke algebra with U_1, U_2 as finite generator and D_{3,1} as infinite generator. In particular, D_{3,1}*D_{3,1}= 6 + 3B_2+ 3C_2 + D_{4,1} and D_{n,1}*D_{3,1}= 3A_{n-3} + 3B_{n-1} + D_{n+1,1} +3D_{n-1,1} up to powers of Z, which inductively gives all D_{n,a}. We use these relations to give a complete description of the Whittaker function of new vectors, our description does not use any realization of representations. Our formulae indicate that in the case of higher primes the Hecke algebra may no longer be finitely generated. This is in progress with Moshe Baruch.
Waldspurger’s theorem has been applied to study the congruent number problem and its several variants. In a project with Jerome Dimabayao we study pi/4 congruent number problem and relate it with recent studies on tilings.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
We computed the relations amongst the basis elements of G modulo K_0(p^2) but we have not yet been able to give the exact set of generators and defining relations for each prime. The relations of the type D_{n,a}*D_{m,b} are much more involved than we expected and seem to no longer give a finite generating set for the Hecke algebra in the case of higher primes. In case, Hecke algebras are not finitely generated, we may no longer be able to describe the Whittaker function by its values on only finitely many double cosets; on the other hand, it may lead to interesting family of infinitely generated Hecke algebras. There has been some recent work where families of infinitely generated Hecke Algebras with infinite presentation are constructed using graph theory.
The research progress has been slightly delayed due to the current Covid situation due to which considerable amount of time is spent on teaching.
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| 今後の研究の推進方策 |
We plan to continue with our study of local Hecke algebra of G modulo K_0(p^2) for general prime p. We will apply relations involving D_{n,a}’s that we obtained for general primes to particular cases of primes to get an idea of possible general picture. In the case p=3, we could get D_{4,1} from D_{3,1}*D_{3,1} and then could get D_{n+1,1} from D_{n,1}*D_{3,1} and so could obtain all the D_{n,1}'s. In the case p=5, using D_{3,a}'s, we get D_{n, a}'s in pairs but haven't been able to find relations that just give D_{n,1}. Here is an example of inductive relation that we get when p=5: for n>3, D_{n,1}*D_{3,1}= 5A_{n-3} + 5B_{n-1}+ 10D_{n-1,3} + 5D_{n-1,4} + 2D_{n+1,2}+D_{n+1,4}. We expect similar picture for other primes. On the other hand, when p=5, we have been able to give complete description of Whittaker functions under certain restriction. We will reinterpret those restrictions in terms of Hecke algebra to get any new information that we can use hopefully to get the complete presentation.
Earlier we used the central elements of the Hecke algebra H(G//K_0(4)) to describe all its finite dimensional representations. For general odd primes, once we have the Hecke algebra presentation, we plan to work out the center and use it to describe all its finite dimensional representations.
On our project on pi/4 congruent number problem, we gave Tunnell-like conditions in terms of certain ternary quadratic forms. We next plan to construct pi/4 congruent triangles by constructing, via Heegner's method, the points of infinite order of corresponding elliptic curves.
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