1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. RAMANUJAN GRAPHS 407 compact subgroup of I1w#oo,v D(Ow)· By strong approximation theory, the above double cosets can ...

408 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS is a principal ideal. Choose the place v to be the place with t as a ...

LECTURE 1. RAMANUJAN GRAPHS 409 Exercise. Show that the eigenvalues of Cay(JF q2, N _ 1 ) are as stated, and prove Proposition 1 ...

410 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS Let S be a K 8 -double coset of GL 2 (1F'q) with cardinality greater ...

LECTURE 1. RAMANUJAN GRAPHS 411 Note that >-x(lFw, a) is nothing but the eigenvalue A4a,xoNFw/F of a Terras graph with base f ...

412 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS two, there is an automorphic form 9b,<,w on D(K)\D(AK )/ D(K 00 ) ...

LECTURE l. RAMANUJAN GRAPHS 413 L(f, s) for some cuspidal newform f of weight 2 for the congruence subgroup I'o(N) with N being ...

414 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS To prove this theorem, one shows that the image of each of the assoc ...

2.1. Spectral theory of hypergraphs Lecture 2. Ramanujan hypergraphs An n-hypergraph X consists of a set of vertices and a set o ...

416 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS (d) (diagonalizability) The operators Ai are simultaneously diagonal ...

LECTURE 2. RAMANUJAN HYPERGRAPHS 417 of (7) vectors w with jSupp(w)I = i. Partition the representatives in S i according to the ...

418 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS Theorem 2.2. [Li, [23]] Given complex numbers z1, ... , Zn with lz1I ...

LECTURE 2. RAMANUJAN HYPERGRAPHS 419 entries are either 1 or n. By applying the above Lemma repeatedly, we m ay write [g] = [M(1 ...

420 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS Observe that fi(x) = x + 1, and, by induction, form 2: 2, fm(d) = L ...

LECTURE 2. RAMANUJAN HYPERGRAPHS 421 2.3. An explicit construction of Ramanujan n-hypergraphs The construction here is parallel ...

422 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS the sum of the invariants of H over the places in S is also zero. De ...

LECTURE 2. RAMANUJAN HYPERGRAPHS 423 (3) Are there automorphic forms for GLn over function fields which are eigen- functions of ...

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Bibliography l. A. Badulescu, Orthogonalite des caracteres pour GLn sur un corps local de caracteristique non nulle, Manuscripta ...

426 W.-C. W. LI, RAMANUJAN GRAPHS AND RAMANUJAN HYPERGRAPHS J.-I. Igusa, Kroneckerian model of fields of elliptic modular funct ...