is completed for primitive characters quickly by establishing the multiplicity-
one principle using Gauss and Ramanujan sums instead of lengthy consid-
erations of inner products. Of course, this is only a special case (the whole
space is spanned by newforms), but it is an important case.
We pay great attention to detail in the subjects of theta functions and
representations by quadratic forms (Chapters 10 and 11) because these are
not sufficiently covered in textbooks, despite having a long history of re-
Because the original notes where written as the course was progressing,
inevitably some redundancy has occurred. Nevertheless we have decided
not to eliminate this redundancy, because it offers the option of selective
reading. For example, our account of the Shimura-Taniyama conjecture for
special curves (the congruent number curves) is self-contained in Chapter 8,
even though one could instead appeal to the later chapters on general theta
Sergei Gelfand, Peter Sarnak and others have convinced me that these
lecture notes might be useful for a large number of graduate students, and
they have urged me to publish them. I would like to thank them for their en-
couragement. I am grateful to W. A. Gonzalez, C. L. Hamer and C. J. Moz-
zochi for helping me in the technical preparation of the original notes. Spe-
cial thanks are expressed to T. Khovanova for corrections and improvements
which she contributed when editing these notes for publication.
Henryk Iwaniec
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