Preface The quadratic Diophantine equations have a quite long and rich history. In the middle of 17-th century, Fermat studied the integers which can be represented by certain binary quadratic forms such as x2 -f y2 and x2 + 2y2. In 19-th century, Gauss laid the foundations of binary quadratic forms by using quadratic fields. He developed the composition theory, genus theory and so on, which was in his famous book " Disquisitions Arithmeticae ". In the beginning of last century, Hilbert proposed his famous 23 problems. Among these problems, he asked to study the arithmetic properties of quadratic forms in his 11-th problem (see [OM2]). Almost at the same time, the p-adic completions of number fields have been developed and the quadratic forms over such fields had been studied. The rational representations of one quadratic forms by another can be solved by the local-global principle (Hasse-Minkowski theorem or Hasse principle) as part of the achievement of the global classfield theory (see [OM1]). However, such local-global principle fails for the integral representations of one quadratic form by another in general. In the first half of last century, the various analytical methods had been developed for studying the integral representations of quadratic forms such as the circle method of Hardy-Littlewood, Hecke's classical modular form theory, Linnik's ergodic theory and Siegel's mass formulae [Si] and so on. Another important progress of quadratic form theory is Witt's geometric lan- guage [Wi] over general fields of characteristic not 2 in the thirties of the last century. By using such language, O'Meara [OM] solved the integral representations of quadratic forms over non-dyadic local fields and 2-adic local fields and Riehm [Ri] gave partial results for general dyadic local fields. In the fifties of last cen- tury, Eichler [Ei] studied quadratic forms from the algebraic group point of view and introduced so called spinor genus theory which became one of the most power- ful tools to study the integral representations of quadratic forms over global fields via the strong approximation theorem for spin groups. Such spinor genus theory was further developed by various effort in [Kn], [Knl], [We], [SP], [HSX] and was eventually complete in [CX] and [Xu3] recently. It is well-known that the theory of quadratic forms (or orthogonal groups ) in characteristic 2 has quite different feature from that of characteristic not 2. Arf [Ar] established the general theory of quadratic forms in characteristic 2 which is the analogy of Witt's work [Wi]. Later on, the integral equivalence for quadratic forms over local fields in characteristic 2 was solved in [Sa]. Riehm [Ril] gave a partial answer to the integral representations of quadratic forms over local fields in characteristic 2. Harder [Ha] established reduction theory for semi-simple algebraic groups over global function fields, which implies the finiteness of class number of a integral quadratic form over a global field of characteristic 2. The relative vii
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