**Book Description:**

*Modular Forms and Special Cycles on Shimura Curves* is a
thorough study of the generating functions constructed from special
cycles, both divisors and zero-cycles, on the arithmetic surface
"M" attached to a Shimura curve "M" over the field of rational
numbers. These generating functions are shown to be the
q-expansions of modular forms and Siegel modular forms of genus two
respectively, valued in the Gillet-Soulé
arithmetic Chow groups of "M". The two types of generating
functions are related via an arithmetic inner product formula. In
addition, an analogue of the classical Siegel-Weil formula
identifies the generating function for zero-cycles as the central
derivative of a Siegel Eisenstein series. As an application, an
arithmetic analogue of the Shimura-Waldspurger correspondence is
constructed, carrying holomorphic cusp forms of weight 3/2 to
classes in the Mordell-Weil group of "M". In certain cases, the
nonvanishing of this correspondence is related to the central
derivative of the standard L-function for a modular form of weight
2. These results depend on a novel mixture of modular forms and
arithmetic geometry and should provide a paradigm for further
investigations. The proofs involve a wide range of techniques,
including arithmetic intersection theory, the arithmetic adjunction
formula, representation densities of quadratic forms, deformation
theory of p-divisible groups, p-adic uniformization, the Weil
representation, the local and global theta correspondence, and the
doubling integral representation of L-functions.

**eISBN:**978-1-4008-3716-8

**Subjects:**Mathematics