By Gérard Laumon, Jean Loup Waldspurger

ISBN-10: 0521470609

ISBN-13: 9780521470605

ISBN-10: 0521470617

ISBN-13: 9780521470612

Cohomology of Drinfeld Modular kinds offers an creation, in volumes, either to this topic and to the Langlands correspondence for functionality fields. it truly is in keeping with classes given by way of the writer who, to maintain the presentation as available as attainable, considers the better case of functionality instead of quantity fields; however, many very important gains can nonetheless be illustrated. numerous appendices on history fabric make this a self-contained e-book. it is going to be welcomed through staff in algebraic quantity thought and illustration conception.

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**Extra resources for Cohomology of Drinfeld Modular Varieties**

**Example text**

The ﬁrst published Cayley graph, appearing in Cayley’s original paper [Ca78], was the Cayley graph for A4 with respect to the generating set {(123), (234)}. Draw this Cayley graph. Let G be the symmetry group of the regular octahedron. a. Compute the order of G. b. Find a set of generators S for G and explain how you know these form a set of generators. ) c. Draw or explain in detail the shape and structure of the Cayley graph ΓG,S . (14) What is the order of the symmetry group of a dodecahedron?

In order to be very concrete, let be thought of as the convex hull of the eight points in R3 whose coordinates are of the form (±1, ±1, ±1), and let Sym( ) consist of all reﬂections and rotations that take back to itself. 34). Let v be the vertex at (1, 1, 1). Then the orbit of v consists of the eight vertices of . Since any element in the stabilizer of v permutes the three edges incident with v, and no non-trivial symmetry ﬁxes these three edges, we have Stab(v) ≈ Sym3 . Thus |G| = |Orb(v)| · |Stab(v)| = 8 · 6 = 48.

47. Follow a procedure similar to the one outlined above to ﬁnd an embedding of the Cayley graph of Dn , with respect to a reﬂection and a rotation through an angle of 2π/n, inside a regular n-gon. You will want to make a judicious choice for the initial point whose orbit corresponds with the elements of Dn . 2 Symmetric Groups At the end of the previous section we produced the Cayley graph of Sym3 with respect to the generating set {(12), (123)}. Here we will examine the Cayley graph of Sym4 with respect to the generating set {(12), (23), (34)}.

### Cohomology of Drinfeld Modular Varieties by Gérard Laumon, Jean Loup Waldspurger

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