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1. CJM 2012 (vol 65 pp. 544)
Iterated Integrals and Higher Order Invariants We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The non-uniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 |
2. CJM 2011 (vol 63 pp. 826)
Singular Moduli of Shimura Curves The $j$-function acts as a parametrization of the classical modular
curve. Its values at complex multiplication (CM) points are called
singular moduli and are algebraic integers. A Shimura curve is a
generalization of the modular curve and, if the Shimura curve has
genus~$0$, a rational parameterizing function exists and when
evaluated at a CM point is again algebraic over~$\mathbf{Q}$. This paper shows
that the coordinate maps given by N.~Elkies for the Shimura
curves associated to the quaternion algebras with discriminants $6$
and $10$ are Borcherds lifts of vector-valued modular forms. This
property is then used to explicitly compute the rational norms of
singular moduli on these curves. This method not only verifies
conjectural values for the rational CM points, but also provides a way
of algebraically calculating the norms of CM points with arbitrarily
large negative discriminant.
Categories:11G18, 11F12 |
3. CJM 2004 (vol 56 pp. 406)
Theta Series, Eisenstein Series and PoincarÃ© Series over Function Fields We construct analogues of theta series, Eisenstein series and
Poincar\'e series for function fields of one variable over finite
fields, and prove their basic properties.
Category:11F12 |
4. CJM 2003 (vol 55 pp. 933)
Renormalized Periods on $\GL(3)$ A theory of renormalization of divergent integrals over torus
periods on $\GL(3)$ is given, based on a relative truncation. It
is shown that the renormalized periods of Eisenstein series have
unexpected functional equations.
Categories:11F12, 11F55 |