233 research outputs found
Rankin-Cohen Type Differential Operators for Siegel Modular Forms
Let H_n be the Siegel upper half space and let F and G be automorphic forms
on H_n of weights k and l, respectively. We give explicit examples of
differential operators D acting on functions on H_n x H_n such that the
restriction of D(F(Z_1) G(Z_2)) to Z = Z_1 = Z_2 is again an automorphic form
of weight k+l+v on H_n. Since the elliptic case, i.e. n=1, has already been
studied some time ago by R. Rankin and H. Cohen we call such differential
operators Rankin-Cohen type operators. We also discuss a generalisation of
Rankin-Cohen type operators to vector valued differential operators.Comment: 19 pages LaTeX2e using amssym.de
Arithmetic and equidistribution of measures on the sphere
Motivated by problems of mathematical physics (quantum chaos) questions of
equidistribution of eigenfunctions of the Laplace operator on a Riemannian
manifold have been studied by several authors. We consider here, in analogy
with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the
Laplace operator on the two dimensional unit sphere which are also
eigenfunctions of an algebra of Hecke operators which act on these spherical
harmonics. We formulate an analogue of the equidistribution of mass conjecture
for these eigenfunctions as well as of the conjecture that their moments tend
to moments of the Gaussian as the eigenvalue increases. For such orthonormal
bases we show that these conjectures are related to the analytic properties of
degree eight arithmetic L-functions associated to triples of eigenfunctions.
Moreover we establish the conjecture for the third moments and give a
conditional (on standard analytic conjectures about these arithmetic
L-functions) proof of the equdistribution of mass conjecture.Comment: 18 pages, an appendix gives corrections to the article "On the
central critical value of the triple product L-function" (In: Number Theory
1993-94, 1-46. Cambridge University Press 1996) by Siegfried Boecherer and
Rainer Schulze-Pillot. Revised version (minor revisions, new abstract), paper
to appear in Communications in Mathematical Physic
Direct Integration of the Topological String
We present a new method to solve the holomorphic anomaly equations governing
the free energies of type B topological strings. The method is based on direct
integration with respect to the non-holomorphic dependence of the amplitudes,
and relies on the interplay between non-holomorphicity and modularity
properties of the topological string amplitudes. We develop a formalism valid
for any Calabi-Yau manifold and we study in detail two examples, providing
closed expressions for the amplitudes at low genus, as well as a discussion of
the boundary conditions that fix the holomorphic ambiguity. The first example
is the non-compact Calabi-Yau underlying Seiberg-Witten theory and its
gravitational corrections. The second example is the Enriques Calabi-Yau, which
we solve in full generality up to genus six. We discuss various aspects of this
model: we obtain a new method to generate holomorphic automorphic forms on the
Enriques moduli space, we write down a new product formula for the fiber
amplitudes at all genus, and we analyze in detail the field theory limit. This
allows us to uncover the modularity properties of SU(2), N=2 super Yang-Mills
theory with four massless hypermultiplets.Comment: 75 pages, 3 figure
Estimating the order of vanishing at infinity of Drinfeld quasi-modular forms
We introduce and study certain deformations of Drinfeld quasi-modular forms
by using rigid analytic trivialisations of corresponding Anderson's t-motives.
We show that a sub-algebra of these deformations has a natural graduation by
the group Z^2 x Z/(q-1)Z and an homogeneous automorphism, and we deduce from
this and other properties multiplicity estimates
A COHOMOLOGICAL INTERPRETATION OF ARCHIMEDEAN ZETA INTEGRALS FOR GL GL (Automorphic forms, Automorphic representations, Galois representations, and its related topics)
This article is a survey on the author's preprint [T. Hara and K. Namikawa, A cohomological interpretation of archimedean zeta integrals for GL3 xGL2, preprint, arXiv:2012.13213.], where the author constructs a p-adic Asai L-functions for irreducible cohomological cuspidal autmorphic representations of GL₂ over CM fields
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