Teaching Statement and Statement of Research Plans - GWDG

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4 Index (with respect to a certain set of generators), is the same as the Cayley graph of a well-known group PGL(d) over a finite field, with respect to a set of easily-described generators. Property (τ), Selberg Property, Ramanujan-Peterson conjecture It is well known that every arithmetic lattice Γ (at least over fields of characteristic 0 and conjecturally always) has the Selberg property. If in addition Γ has the congruence subgroup property, then Γ has also (τ). This leads to following conjecture: see [16] for more detail. Conjecture 1. Let Γ be a lattice in a semisiple group. congruence subgroup property if and only if Γ has (τ). Then Γ has the There is a close relationship between the congruence subgroup property, the Selberg property and the Ramanujan-Peterson conjecture. One of my main goals is to study these relationships more deeply with the viewpoint of obtaining more information related to the above conjecture. I have some ideas how to generalise the known case to the higher dimensional case and I am interested in arithmetic groups having property (τ) over a division ring and I want to consider the effect of the (τ) property on the lifting of the representations to the similar corresponding linear groups such as GL or SL under the Langlands corespondence. Group action on the set of galleries of a Bruhat-Tits Building Let D be a division algebra over a function field K i.e, a finite extension of F q (t) with dim D K = d 2 , and π = (π) a place which is unramified in D, i.e, D π := D ⊗ K K π ∼ = Md (K π ) and π is an irreducible polynomial in OsupsetF q [t] . Let M 0 and M π 0 be the set of maximal orders in D and D π respectively. We can see first that,M π 0 can be interprated as the set of vertices of a Bruhat- Tits building X π ∗ associated to PGL d (K π ). We can also consider another Bruhat-Tits building X (global case) with M 0 as the set of vertices. Let R be a fix maximal order in D. Now consider the set: The unit group Γ (1) R X 0 (R) π = {M ∈ M 0 | M v = R v for all v ≠ π} R is equal to the number of orbits of Γ (1) R := R ⊗ O[ 1 π ]× acts on X 0 (R) π and the class number of under this action. For any oder S
Index 5 of level π n in D, we denote its nth associated unit group with U (n) S := (S ⊗ O[ 1 π ])× . I will generalise the work of three “Son” (Rosson, H J. and Ellison,B J. and Wilson,J B.), TREES, HECKE OPERATORS, AND QUADRATIC FORMS, http://darkwing.uoregon.edu/ jwilson7/math/PrePrintTree.pdf, 2001 as follows: I will show there is a group action of U (n) S := (S ⊗O[ 1 π ])× on the set of galeries of length n on the Bruhat-Tits building X ∗ which gives us the class number of S as the number of orbits of U (n) S := (S ⊗ O[ 1 π ])× under this action. Also I will study Brandt matrices and show that there is some recursive formulas to compute these matrices and also to compute the number of solutions arised from reduced norm equation. Bruhat-Tits Building and the Zeta functions of nilpotent groups Using the combinatorial geometry of the Bruhat-Tits building for SL(d,Q p ), C. Voll introduced a new method to compute local normal zeta functions of nilpotent groups (see [24]). This motivates me to study more the nature of the Bruhat-Tits Building arising in my thesis for the same situation but over the local function field instead of Q p in a more technical point of view. Zeta and (L)-functions of finite graphs and finite regular-hypergraphs Ihara obtained, for a discrete co-compact and torsion-free subgroup Γ of SL(2,Q d ) and the associated Bruhat-Tits tree (X . ), a zeta function of the finite graph Γ\(X ∗ ). Ihara, using the theory of automorphic forms, has presented successfully his beautiful discovery: the identity of the zeta function of the finite graph Γ\(X ∗ ) and the zeta function of certain Shimura curve S Γ reduced modulo a prime p (see [10], [9]). One of the main ingredients required, to generalise the above connection to higher-rank reductive groups G over a global field K, is to obtain a suitable definition for the zeta function of a finite hypergraph (= a finite quotient of the Bruhat-Tits building) associated to the G(K ν ) for a place ν. There are some ideas by C.Ballantine and W. Li and A.Deitamr (based on Selberg’s trace formula) in the higher dimensional case. But it seems that a thorough structured theory could be worked out. In my recent work [22], I have three main goals: 1) Thanks to Garland’s work [6] I define a Zeta function related to the closed
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Page 9: Index 9 [17] Ronan. M. Buildings: M

Teaching Statement and Statement of Research Plans - GWDG

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