We give a new proof of Serre’s result that a Noetherian local ring is regular if and only if it has finite global dimension. Our proof avoids the explicit construction of a Koszul complex.
Given a formula for the Mahler measure of a rational function expressed in terms of polylogarithms, we describe a new method that allows us to construct a rational function with 2 more variables and whose Mahl...
In 2013, Lemke Oliver classified all eta-quotients which are theta functions. In this paper, we unify the eta–theta functions by constructing mock modular forms from the eta–theta functions with even character...
Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal ...
Let d≥1 be fixed. Let F be a number field of degree d, and let E/F be an elliptic curve. Let E(F)_{tors} be the torsion subgroup of E(F). In 1996, Merel proved the uniform boundedness conjecture, i.e., there is a co...
We exploit transformations relating generalized q-series, infinite products, sums over integer partitions, and continued fractions, to find partition-theoretic formulas to compute the values of constants such as
We study the distribution of rational points of bounded height on a one-sided equivariant compactification of PGL_{2} using automorphic representation theory of PGL_{2}.
The smallest part is a rational function. This result is similar to the closely related case of partitions with fixed differences between largest and smallest parts which has recently been studied through anal...
We show that a certain family of the coefficients of a Drinfeld-Goss modular form with certain power eigenvalues for the Hecke operators at degree 1 primes the can be expressed as polynomial multiples of the f...
In 1982, Gessel showed that the Apéry numbers associated to the irrationality of ζ(3) satisfy Lucas congruences. Our main result is to prove corresponding congruences for all known sporadic Apéry-like sequences. ...
A interpretation of the Rogers–Zudilin approach to the Boyd conjectures is established. This is based on a correspondence of modular forms which is of independent interest. We use the reinterpretation for two ...
Authors: Nikolaos Diamantis, Michael Neururer and Fredrik Strömberg
We discuss the mean values of multiplicative functions over function fields. In particular, we adapt the authors’ new proof of Halász’s theorem on mean values to this simpler setting. Several of the technical ...
Authors: Andrew Granville, Adam J. Harper and Kannan Soundararajan
We use a generalized Lambert series identity due to the first author to present q-series proofs of recent results of Imamoğlu, Raum and Richter concerning recursive formulas for the coefficients of two 3rd order ...
Authors: Song Heng Chan, Renrong Mao and Robert Osburn
We revisit Huber’s theory of continuous valuations, which give rise to the adic spectra used in his theory of adic spaces. We instead consider valuations which have been reified, i.e., whose value groups have bee...
The generating function of partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part is shown to be the third order mock theta function ω(q) (resp. ν(−q)). Simil...
Authors: George E. Andrews, Atul Dixit and Ae Ja Yee
We prove that the Heegner cycles of codimension m+1 inside Kuga-Sato type varieties of dimension 2m+1 are coefficients of modular forms of weight 3/2+m in the appropriate quotient group. The main technical tool f...
We study the leading digit laws for the matrix entries of a linear Lie group G. For non-compact G, these laws generalize the following observations: (1) the normalized Haar measure of the Lie group
Mock modular forms, which give the theoretical framework for Ramanujan’s enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular para...
Authors: Claudia Alfes, Michael Griffin, Ken Ono and Larry Rolen
In 1919, P. A. MacMahon studied generating functions for generalized divisor sums. In this paper, we provide a framework in which to view these generating functions in terms of Jacobi forms, and prove that the...
In a recent paper, Rose proves that certain generalized sum-of-divisor functions are quasi-modular forms for some congruence subgroup and conjectures that these forms are quasi-modular for Γ
...
We construct a natural basis for the space of weak harmonic Maass forms of weight 5/2 on the full modular group. The nonholomorphic part of the first element of this basis encodes the values of the ordinary pa...
Recently, Bruinier and Ono found an algebraic formula for the partition function in terms of traces of singular moduli of a certain non-holomorphic modular function. In this paper we prove that the rational po...
The aim of this paper is to study the convergence and divergence of the Rogers-Ramanujan and the generalized Rogers-Ramanujan continued fractions on the unit circle. We provide an example of an uncountable set...
Authors: Emil-Alexandru Ciolan and Robert Axel Neiss
In a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using extended Hall-Littlewood polynomials P_{
λ
}(x
...
Given an elliptic curve E and a prime p of (good) supersingular reduction, we formulate p-adic analogues of the Birch and Swinnerton-Dyer conjecture using a pair of Iwasawa functions L_{
...}
Inspired by previous work of Bruinier-Ono and Mertens-Rolen, we study class polynomials for non-holomorphic modular functions arising from modular forms of negative weight.
Authors: Joschka J. Braun, Johannes J. Buck and Johannes Girsch
Riemann’s non-differentiable function and Gauss’s quadratic reciprocity law have attracted the attention of many researchers. Here we provide a combined proof of both the facts. In (Proc. Int. Conf.–NT 1;107–1...
Authors: Kalyan Chakraborty, Shigeru Kanemitsu and Li Hai Long
We establish a Kronecker limit formula for the zeta function ζ_{
F
}(s,A) of a wide ideal class A of a totally real number field F of degree n. This formula...
Using the Ratios Conjecture as introduced by Conrey, Farmer and Zirnbauer, we obtain closed formulas for the one-level density for two families of L-functions attached to elliptic curves, and we can then deter...
Authors: Chantal David, Duc Khiem Huynh and James Parks
We establish Sturm bounds for degree g Siegel modular forms modulo a prime p, which are vital for explicit computations. Our inductive proof exploits Fourier-Jacobi expansions of Siegel modular forms and properti...
Authors: Olav K Richter and Martin Westerholt-Raum
For X a curve over a field of positive characteristic, we investigate when the canonical representation of Aut(X) on H^{
0
}(X,Ω_{X}) is irreducible. Any curve with an...
Authors: Benjamin Gunby, Alexander Smith and Allen Yuan
We use the theory of congruences between modular forms to prove the existence of newforms with square-free level having a fixed number of prime factors such that the degree of their coefficient fields is arbit...
Authors: Luis Víctor Dieulefait, Jorge Jiménez Urroz and Kenneth Alan Ribet