arXiv subjects
Publications and source records attributed to Tian An Wong.
We generalize Merel's work on universal Fourier expansions to Bianchi modular forms over Euclidean imaginary quadratic fields, under the assumption of the nondegeneracy of a pairing between Bianchi modular forms and Bianchi modular symbols. Among the key inputs is a computation of the action of Hecke operators on Manin symbols, and building upon the Heilbronn-Merel matrices constructed by Mohamed.
We express the discrete noncuspidal terms in the spectral side of the trace formula for GL(2) in terms of orbital integrals, obtaining a geometric expansion for the cuspidal part of the trace formula. Assuming the Ramanujan conjecture for GL(2), this is equal to the tempered part of the trace formula, providing an alternate approach to the method of Frenkel-Langlands-Ngo in the context of Beyond Endoscopy. Using this, we establish a formula which in principle leads to an $r$-trace formula, and conclude with some remarks regarding the primitivization of the trace formula.
The continuous spectrum to the spectral side of the Arthur-Selberg trace formula is described in terms of intertwining operators, whose normalising factors involve quotients of $L$-functions. In this paper, we derive two expressions in the case of SL(2) over a number field in terms of the Riemann-Weil explicit formula: as a sum over zeroes of the associated $L$-functions, and as a sum of adelic distributions on Weil groups. As an application, we obtain an expression for a lower bound for the sums over zeroes with respect to the truncation parameter for Eisenstein series.
We introduce a refinement of the classical Liouville function to primes in arithmetic progressions. Using this, we discover new biases in the appearances of primes in a given arithmetic progression in the prime factorizations of integers. For example, we observe that the primes of the form $4k+1$ tend to appear an even number of times in the prime factorization of a given integer, more so than for primes of the form $4k+3$. We are led to consider variants of Pólya's conjecture, supported by extensive numerical evidence, and its relation to other conjectures.
We extend the work of S. Zhang and Yuan-Zhang-Zhang to obtain a Gross-Zagier formula for modular forms of even weight in terms of an arithmetic intersection pairing of CM-cycles on Kuga-Sato varieties over Shimura curves. Combined with a result of the first author and de Vera-Piquero adapting Kolyvagin's method of Euler systems to this setting, we bound the associated Selmer and Tate-Shafarevich groups, assuming the non-vanishing of the derivative of the L-function at the central point.
We outline a general algorithm for verifying whether a subset of the integers is a more sum than differences (MSTD) set, also known as sum dominated sets, and give estimates on its computational complexity. We conclude with some numerical results on large MSTD sets and MSTD subsets of $[1,N]\cap \mathbb Z$ for $N$ up to 160.
We study the Ehrhart theory of quadratic irrational polytopes that undergo vector dilations. That is, for a given polytope with vertices in $\mathbb{Q}(\sqrt{D})$, and a different dilation factor for each facet, we show that the leading term of the lattice-point count behaves similar to an Ehrhart polynomial, generalizing previous work of Borda on scalar dilations of quadratic irrational polytopes. As a result, a form of the Ehrhart-Macdonald reciprocity law is obtained for the leading term.
We introduce a variation on the prime divisor function $B(n)$ of Alladi and Erdős, a close relative of the sum of proper divisors function $s(n)$. After proving some basic properties regarding these functions, we study the dynamics of its iterates and discover behaviour that is reminiscent of aliquot sequences. We prove that no unbounded sequences occur, analogous to the Catalan-Dickson conjecture, and give evidence towards the analogue of the Erdős-Granville-Pomerance-Spiro conjecture on the pre-image of $s(n)$.
Recent work of Altuğ continues the preliminary analysis of Langlands' Beyond Endoscopy proposal for $GL(2)$ by removing the contribution of the trivial representation to the trace formula using a Poisson summation formula. We show that Altuğ's method of smoothing real elliptic orbital integrals by an approximate functional equation extends to $GL(n)$. We also discuss the case of an arbitrary reductive group, and remaining obstructions for applying Poisson summation.