By Joseph Bernstein, Stephen Gelbart, S.S. Kudla, E. Kowalski, E. de Shalit, D. Gaitsgory, J.W. Cogdell, D. Bump

This e-book offers a wide, easy creation to the Langlands application, that's, the speculation of automorphic varieties and its reference to the idea of L-functions and different fields of arithmetic. all of the twelve chapters makes a speciality of a selected subject dedicated to detailed circumstances of this system. The ebook is appropriate for graduate scholars and researchers.

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**Example text**

When a maximal point or minimal point occurs on the interior of a double point arc, this is a type II Reidemeister move. The pair of strings involved has no fold lines. In the chart this is represented by a critical point of a solid arc. 3. When an isolated triple point occurs among three double point arcs and there are three sheets of surface intersecting pair wise along these arcs, this is called a type III Reidemeister move. The three sheets involved have no fold lines. In the chart this is represented by a 6-valent vertex among 6 solid arcs.

The fact that each of these integers is odd reflects the fact that the trefoil depicted is left-handed. Associated to a knotted surface diagram we obtain a sequence of Gauss codes; the elements in the sequence are the codes associated to classical diagrams that are planar intersections with the surface. In case the classical diagrams are link diagrams each component has a sequence of codes associated to it. We read the sequence of Gauss codes from the top to the bottom of the knotted surface where top and bottom are measured with respect to the fixed height function.

4) The projection of a point at which an arc of double points crosses a fold is a 4-valent vertex. Two of the edges at this vertex are solid; the other two are dotted. A circle in the retinal plane that encompasses such a vertex encounters the edges in the cyclic order (solid, solid, dotted, dotted). (5) The points of the retinal plane at which the double points cross are 4-valent vertices at which all of the incoming edges are solid. (6) The points of the retinal plane at which the fold lines cross are 4-valent vertices at which all of the incoming edges are dotted.