arXiv subjects
Publications and source records attributed to Tian An Wong.
Using analytic number theory techniques, Altuğ showed that the contribution of the trivial representation to the Arthur-Selberg trace formula for GL(2) over $\Q$ could be cancelled by applying a modified Poisson summation formula to the regular elliptic contribution. Drawing on recent works, we re-examine these methods from an adelic perspective.
We construct stable geometric and spectral transfer factors for a general reductive group and develop some of their basic properties, assuming the refined local Langlands correspondence. Using our definition of stable geometric transfer factors, we show that the stable transfer conjecture for orbital integrals implies the stable transfer of characters and vice versa. The latter is also implied by local Langlands, and in particular this establishes archimedean stable geometric transfer. Finally, we show how the stable geometric transfer factors can be used to define stable spectral transfer factors.
Using an explicit Eichler-Shimura-Harder isomorphism, we establish the analogue of Manin's rationality theorem for Bianchi periods and hence special values of $L$-functions of Bianchi cusp forms. This gives a new short proof of a result of Hida in the case of Euclidean imaginary quadratic fields. In particular, we give an explicit proof using the space of Bianchi period polynomials constructed by Karabulut and describe the action of Hecke operators.
We establish a doubly-weighted vertical Sato-Tate law for GL(4) with explicit error terms. The main ingredient is an extension of the orthogonality relation for Maass cusp forms on GL(4) of Goldfeld, Stade, and Woodbury from spherical to general forms, and without their assumption of the Ramanujan conjecture for the error term.
We construct a generalization of the Dedekind-Rademacher cocycle to congruence subgroups of $\mathrm{SL}_2(\mathbb C)$, and derive some of its basic properties. In particular, we show that it parametrizes a family of $L$-values and prove the integrality of these values.
We show that the graph of normalized elliptic Dedekind sums is dense in its image for arbitrary imaginary quadratic fields, generalizing a result of Ito in the Euclidean case. We also derive some basic properties of Martin's continued fraction algorithm for arbitrary imaginary quadratic fields.
We express the contribution of certain maximal parabolic Eisenstein series to the spectral side of the Arthur--Selberg trace formula for GL$(n)$ in terms of zeroes of Rankin--Selberg $L$-functions, generalizing previous results for GL(2) and Hecke $L$-functions. As applications, we prove a lower bound for the sum of these zeroes, and a base change relation between the zeroes in the case $n=2$ for cyclic extensions of prime degree.
In the early 2000's, R. Langlands proposed a strategy called Beyond Endoscopy to attack the principle of functoriality, which is one of the central questions of present day mathematics. A first step was achieved by A. Altug who worked with the group $GL(2,\mathbb{Q})$ in a series of three papers. We generalize the first part of this result to a class of totally real number fields. In particular, we cancel the contribution of the trivial and special representations to the trace formula using an additive Poisson summation on the regular elliptic terms.
We show that the values of elliptic Dedekind sums, after normalization, are equidistributed mod 1. The key ingredient is a non-trivial bound on generalized Selberg-Kloosterman sums for discrete subgroups of $\PSL_2(\mathbb C)$ using Poincaré series.
We prove a first Kronecker limit formula for cofinite discrete subgroups of SL$(2,\mathbb{C})$, also called Kleinian groups, generalizing a method of Goldstein over SL$(2,\mathbb R)$. The proof uses the Fourier expansion of Eisenstein series, leading to an analogue of the logarithm of the Dedekind eta function, and whose transformation law produces a function analogous to one related to Dedekind sums. We derive several basic properties of this new function.
We consider a generalization of the classical 100 Prisoner problem and its variant, involving empty boxes, whereby winning probabilities for a team depend on the number of attempts, as well as on the number of winners. We call this the unconstrained 100 prisoner problem. After introducing the 3 main classes of strategies, we define a variety of `hybrid' strategies and quantify their winning-efficiency. Whenever analytic results are not available, we make use of Monte Carlo simulations to estimate with high accuracy the winning-probabilities. Based on the results obtained, we conjecture that all strategies, except for the strategy maximizing the winning probability of the classical (constrained) problem, converge to the random strategy under weak conditions on the number of players or empty boxes. We conclude by commenting on the possible applications of our results in understanding processes of information retrieval, such as ``memory'' in living organisms.
Elliptic Dedekind sums were introduced by R. Sczech as generalizations of classical Dedekind sums to complex lattices. We show that for any lattice with real $j$-invariant, the values of suitably normalized elliptic Dedekind sums are dense in the real numbers. This extends an earlier result of Ito for Euclidean imaginary quadratic rings. Our proof is an adaptation of the recent work of Kohnen, which gives a new proof of the density of values of classical Dedekind sums.
We establish endoscopic and stable trace formulas whose discrete spectral terms are weighted by automorphic $L$-functions, by the use of basic functions that are incorporated into the global spectral and geometric coefficients. This is a continuation of a previous work of the author which established the corresponding weighted invariant trace formula for noncompactly supported test functions. The meromorphic continuation of these weighted trace formulas would yield $r$-trace formulas, and can therefore be seen as precursors to them. Along the way, we formulate a weighted form of the Langlands-Shelstad transfer conjecture, generalizing the weighted fundamental lemma of Arthur.
The Three Gap Theorem, also known as the Steinhaus Conjecture, is a classical result on the combinatorics of the fractional part function, and has since been generalized in many ways. In this paper, we pose a new problem related to these results: for which other periodic functions does an analogue of the Three Gap Theorem hold? We prove analogous results for certain classes of piecewise-linear periodic functions and demonstrate the existence of functions for which no bound exists on the number of gap lengths.
This paper begins a new approach to the $r$-trace formula, without removing the nontempered contribution to the spectral side. We first establish an invariant trace formula whose discrete spectral terms are weighted by automorphic $L$-functions. This involves extending the results of Finis, Lapid, and Müller on the continuity of the coarse expansion of Arthur's noninvariant trace formula to the refined expansion, and then to the invariant trace formula, while incorporating the use of basic functions at unramified places.
We discuss mock automorphic forms from the point of view of representation theory, that is, obtained from weak harmonic Maass forms give rise to nontrivial $(\mathfrak{g},K)$-cohomology. We consider the possibility of replacing the `holomorphic' condition with `cohomological' when generalizing to general reductive groups. Such a candidate allows for growing Fourier coefficients, in contrast to automorphic forms under the Miatello-Wallach conjecture. In the second part, we provide an overview of the connection with BPS black hole counts as a physical motivation for studying mock automorphic forms.
We generalize Sczech's Eisenstein cocycle for $\mathrm{GL}(n)$ over totally real extensions of $\mathbb{Q}$ to finite extensions of imaginary quadratic fields. By evaluating the cocycle on certain cycles, we parametrize complex values of Hecke L-functions previously considered by Colmez, giving a cohomological interpretation of his algebraicity result on special values of the L-functions.
S.D. Miller and F. Zhou have proved a balanced Voronoi summation formula for GL$_N$ over $\mathbb Q$, which allows one to control the dimensions of the Kloosterman sums appearing on either side of the Voronoi formula. In this note, we prove a balanced Voronoi formula over an arbitrary number field, starting with the Voronoi summation formula of A. Ichino and N. Templier over number fields, allowing one to extend recent results on spectral reciprocity laws to number fields, in special cases.