Related papers: Eisenstein series on affine Kac-Moody groups over …

In this text we generalize the classical Jacobi Eisenstein series as they were discussed by Eichler and Zagier to arbitrary lattices. We use two different methods to derive the general Fourier expansion. The last two sections give formulas…

We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series…

Eisenstein series are real analytic functions which play a central role in spectral theory of the hyperbolic Laplacian. Kronecker limit formulas determine their connection to modular forms. The main result of this work is Theorem 7.2 in…

In the former part of this paper, we summarize our previous results on infinite series involving the hyperbolic sine function, especially, with a focus on the hyperbolic sine analogue of Eisenstein series. Those are based on the classical…

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}_2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv…

In this paper we prove a hybrid subconvexity bound for class group $L$-functions associated to a quadratic extension $K/\mathbb{Q}$ (real or imaginary). Our proof relies on relating the class group $L$-functions to Eisenstein series…

For an almost split Kac-Moody group G over a local non-archimedean field, the last two authors constructed a spherical Hecke algebra H (over the complex numbers C, say) and its Satake isomorphism with the commutative algebra of Weyl…

Given a periodic function $f$, we study the almost everywhere and norm convergence of series $\sum_{k=1}^\infty c_k f(kx)$. As the classical theory shows, the behavior of such series is determined by a combination of analytic and number…

For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when…

We derive explicit formulas for some Kloosterman sums on $\Gamma_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.

We develop $L$-functions ratios conjecture with one shift in the numerator and denominator in certain ranges for the family of cubic Hecke $L$-functions of prime moduli over the Eisenstein field using multiple Dirichlet series under the…

We construct a map from the classifying space of a discrete Kac-Moody group over the algebraic closure of the field with p elements to the classifying space of a complex topological Kac-Moody group and prove that it is a homology…

We study a class of newly-introduced CFTs associated with even quadratic forms of general signature, which we call generalized Narain theories. We first summarize the properties of these theories. We then consider orbifolds of these…

We research the location of the zeros of the Eisenstein series and the modular functions from the Hecke type Faber polynomials associated with the normalizers of congruence subgroups which are of genus zero and of level at most twelve. In…

In 1980 Lusztig proved a stabilisation property of the affine Kazhdan-Lusztig polynomials. In this paper we give a categorical version of such a result using the theory of sheaves on moment graphs. This leads us to associate with any…

We provide for primes $p\ge 5$ a method to compute valuations appearing in the "formal" Katz expansion of the family $\frac{E_{k}^{\ast}}{V(E_{k}^{\ast})}$ derived from the family of Eisenstein series $E_{k}^{\ast}$. The overconvergence…

For a split Kac-Moody group (in J. Tits' definition) over a field endowed with a real valuation, we build an ordered affine hovel on which the group acts. This construction generalizes the one already done by S. Gaussent and the author when…

For any rational prime $p$, we define a certain $p$-stabilization of holomorphic Siegel Eisenstein series for the symplectic group $\text{Sp}(2n)_{/\mathbb{Q}}$ of an arbitrary genus $n \ge 1$. In addition, we derive an explicit formula for…

Let G be a finite group. We systematically exploit general homological methods in order to reduce the computation of G-equivariant KK-theory to topological equivariant K-theory. The key observation is that the functor assigning to a…

We generalise the sum-of-divisors-function $\sigma$ and evenness to the rings of integers of certain algebraic number fields. In particular, we present necessary and sufficient conditions for even Eisenstein integers to be (norm-)perfect…