99 research outputs found
Traces of CM values of modular functions
Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the theta correspondence, we generalize this result to traces of CM values of (weakly holomorphic) modular functions on modular curves of arbitrary genus. We also study the theta lift for the weight 0 Eisenstein series for SL2() and realize a certain generating series of arithmetic intersection numbers as the derivative of Zagier's Eisenstein series of weight 3/2. This recovers a result of Kudla, Rapoport and Yang
Degenerate Whittaker functions for Sp_n(R)
In this paper, we construct Whittaker functions with exponential growth for the degenerate principal series of the symplectic group of genus n induced from the Siegel parabolic subgroup. This is achieved by explicitly constructing a certain Goodman–Wallach operator which yields an intertwining map from the degenerate principal series to the space of Whittaker functions, and by evaluating it on weight- ℓ standard sections. We define a differential operator on such Whittaker functions which can be viewed as generalization of the ξ -operator on harmonic Maass forms for \SL2(\R)
On the injectivity of the Kudla-Millson lift and surjectivity of the Borcherds lift
We consider the Kudla-Millson lift from elliptic modular forms of weight (p+q)/2 to closed q-forms on locally symmetric spaces corresponding to the orthogonal group O(p,q). We study the L²-norm of the lift following the Rallis inner product formula. We compute the contribution at the Archimedian place. For locally symmetric spaces associated to even unimodular lattices, we obtain an explicit formula for the L²-norm of the lift, which often implies that the lift is injective. For O(p,2) we discuss how such injectivity results imply the surjectivity of the Borcherds lift
A -adic Approach to the Weil Representation of Discriminant Forms Arising from Even Lattices
Suppose that is an even lattice with dual and level . Then the
group , which is the unique non-trivial double cover of
, admits a representation , called the Weil
representation, on the space . The main aim of this paper
is to show how the formulae for the -action of a general element of
can be obtained by a direct evaluation which does not
depend on ``external objects'' such as theta functions. We decompose the Weil
representation into -parts, in which each -part can be seen as
subspace of the Schwartz functions on the -adic vector space
. Then we consider the Weil representation of
on the space of Schwartz functions on
, and see that restricting to just
gives the -part of again. The operators attained by the Weil
representation are not always those appearing in the formulae from 1964, but
are rather their multiples by certain roots of unity. For this, one has to find
which pair of elements, lying over a matrix in , belong
to the metaplectic double cover. Some other properties are also investigated.Comment: 29 pages, shortened a lo
Correspondences in Arakelov geometry and applications to the case of Hecke operators on modular curves
In the context of arithmetic surfaces, Bost defined a generalized Arithmetic
Chow Group
(ACG) using the Sobolev space L^2_1. We study the behavior of these groups
under pull-back and push-forward and we prove a projection formula.
We use these results to define an action of the Hecke operators on the ACG of
modular curves and to show that they are self-adjoint with respect to the
arithmetic intersection product. The decomposition of the ACG in
eigencomponents which follows allows us to define new numerical invariants,
which are refined versions of the self-intersection of the dualizing sheaf.
Using the Gross-Zagier formula and a calculation due independently to Bost and
Kuehn we compute these invariants in terms of special values of L series. On
the other hand, we obtain a proof of the fact that Hecke correspondences acting
on the Jacobian of the modular curves are self-adjoint with respect to the
N\'eron-Tate height pairing.Comment: 38 pages. Minor correction
- …