We show that there exists a sequence of genus three curves defined over the rationals in which the height of a canonical Gross–Schoen cycle tends to infinity.
We show that certain products of Whittaker functions and Schwartz functions on a general linear group extend to Whittaker functions on a larger general linear group. This generalizes results of Cogdell and Pia...
The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions...
Authors: Clemens Fuchs, Christoph Hutle and Florian Luca
An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams
For any hyperelliptic curve X, we give an explicit basis of the first de-Rham cohomology of X in terms of Čech cohomology. We use this to produce a family of curves in characteristic
...
We give a complete description of the arboreal Galois representation of a certain postcritically finite cubic polynomial over a large class of number fields and for a large class of basepoints. This is the fir...
Authors: Robert L. Benedetto, Xander Faber, Benjamin Hutz, Jamie Juul and Yu Yasufuku
In 2000 Iwaniec, Luo, and Sarnak proved for certain families of L-functions associated to holomorphic newforms of square-free level that, under the Generalized Riemann Hypothesis, as the conductors tend to infini...
Authors: Owen Barrett, Paula Burkhardt, Jonathan DeWitt, Robert Dorward and Steven J. Miller
We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a r...
We introduce the notion of primitive elements in arbitrary truncated p-divisible groups. By design, the scheme of primitive elements is finite and locally free over the base. Primitive elements generalize the “po...
For a given generalized eta-quotient, we show that linear progressions whose residues fulfill certain quadratic equations do not give rise to a linear congruence modulo any prime. This recovers known results f...
In recent work, Bacher and de la Harpe define and study conjugacy growth series for finitary permutation groups. In two subsequent papers, Cotron, Dicks, and Fleming study the congruence properties of some of ...
We study the problem of finding the Artin L-functions with the smallest conductor for a given Galois type. We adapt standard analytic techniques to our novel situation of fixed Galois type and obtain much improve...
In this paper, we evaluate the Faltings height of an elliptic curve with complex multiplication by an order in an imaginary quadratic field in terms of Euler’s Gamma function at rational arguments.
Authors: Adrian Barquero-Sanchez, Lindsay Cadwallader, Olivia Cannon, Tyler Genao and Riad Masri
We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations ...
A conjecture connected with quantum physics led N. Katz to discover some amazing mixed character sum identities over a field of q elements, where q is a power of a prime
...
We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring
...
We prove an explicit formula for the polynomial part of a restricted partition function, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomial...
We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Dèbes and Emsalem can be used to prove this statement in the...
We generalize a result of Garvan on inequalities and interpretations of the moments of the partition rank and crank functions. In particular for nearly 30 Bailey pairs, we introduce a rank-like function, estab...
Authors: Catherine Babecki, Chris Jennings-Shaffer and Geoffrey Sangston
In a recent paper, Bacher and de la Harpe study the conjugacy growth series of finitary permutation groups. In the course of studying the coefficients of a series related to the finitary alternating group, the...
Authors: Tessa Cotron, Robert Dicks and Sarah Fleming
Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko’s method for exhibiting irregularity in the gaps between consecutive zeros of
...
The mock theta conjectures are ten identities involving Ramanujan’s fifth-order mock theta functions. The conjectures were proven by Hickerson in 1988 using q-series methods. Using methods from the theory of harm...
In this series we examine the calculation of the 2kth moment and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. The present paper begi...
In Pollack and Stevens (Ann Sci Éc Norm Supér 44(1):1–42, 2011), efficient algorithms are given to compute with overconvergent modular symbols. These algorithms then allow for the fast computation of p-adic L-fun...
Authors: Evan P. Dummit, Márton Hablicsek, Robert Harron, Lalit Jain, Robert Pollack and Daniel Ross
We revisit the mathematics that Ramanujan developed in connection with the famous “taxi-cab” number 1729. A study of his writings reveals that he had been studying Euler’s diophantine equation