Number Theory

Refinements of Artin's primitive root conjecture

Speaker: 
Paul Peringuey
Date: 
Mon, Mar 3, 2025
Location: 
PIMS, University of Lethbridge
Online
Zoom
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

Let $\rm{ord}_p(a)$ be the order of $a$ in $( \mathbb{Z} / p \mathbb{Z} )^*$. In 1927, Artin conjectured that the set of primes $p$ for which an integer $a\neq -1,\square$ is a primitive root (i.e. $\rm{ord}_p(a)=p-1$) has a positive asymptotic density among all primes. In 1967 Hooley proved this conjecture assuming the Generalized Riemann Hypothesis (GRH). In this talk we will study the behaviour of $\rm{ord}_p(a)$ as $p$ varies over primes, in particular we will show, under GRH, that the set of primes $p$ for which $\rm{ord}_p(a)$ is “$k$ prime factors away” from $p-1$ has a positive asymptotic density among all primes except for particular values of $a$ and $k$. We will interpret being “$k$ prime factors away” in three different ways, namely $k=\omega(\frac{p-1}{\rm{ord}_p(a)})$, $k=\Omega(\frac{p-1} {\rm{ord}_p(a)})$ and $k=\omega(p-1)-\omega(\rm{ord}_p(a))$, and present conditional results analogous to Hooley's in all three cases and for all integer $k$. From this, we will derive conditionally the expectation for these quantities. Furthermore we will provide partial unconditional answers to some of these questions. This is joint work with Leo Goldmakher and Greg Martin.

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Euler products inside the critical strip

Speaker: 
Arshay Sheth
Date: 
Tue, Feb 25, 2025
Location: 
Online
Zoom
Abstract: 

Even though Euler products of L-functions are generally valid only to the right of the critical strip, there is a strong sense in which they should persist even inside the critical strip. Indeed, the behaviour of Euler products inside the critical strip is very closely related to several major problems in number theory including the Riemann Hypothesis and the Birch and Swinnerton-Dyer conjecture. In this talk, we will give an introduction to this topic and then discuss recent work on establishing asymptotics for partial Euler products of L-functions in the critical strip. We will end by giving applications of these results to questions related to Chebyshev's bias.

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Perfect powers as sum of consecutive powers

Speaker: 
Lucas Villagra Torcomian
Date: 
Mon, Feb 24, 2025
Location: 
Zoom
Online
Conference: 
Lethbridge Number Theory and Combinatorics Seminar
Abstract: 

In 1770 Euler observed that $3^3 + 4^3 + 5^3 = 6^3$ and asked if there was another perfect power that equals the sum of consecutive cubes. This captivated the attention of many important mathematicians, such as Cunningham, Catalan, Genocchi and Lucas. In the last decade, the more general equation $x^k + (x+1)^k + \cdots + (x+d)^k = y^n$ began to be studied. In this talk we will focus on this equation. We will see some known results and one of the most used tools to attack this kind of problems. At the end we will show some new results that appear in arXiv:2404.03457.

Class: 

Bad reduction of rational maps

Speaker: 
Matt Olechnowicz
Date: 
Thu, Feb 13, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

We show that the reduction of a projective endomorphism modulo a discrete valuation naturally takes the form of a set-theoretic correspondence. This raises the possibility of classifying "reduction types" of such dynamical systems, reminiscent of the additive/multiplicative dichotomy for elliptic curves. These correspondences facilitate the exact evaluation of certain integrals of dynamical Green's functions, which arise as local factors in the context of counting rational points ordered by the Call-Silverman canonical height. No prior knowledge of arithmetic dynamics will be assumed.

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Zeta functon of F-gauges and special values

Speaker: 
Shubhodip Mondal
Date: 
Thu, Feb 6, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

In 1966, Tate proposed the Artin–Tate conjectures, which expresses special values of zeta function associated to surfaces over finite fields. Conditional on the Tate conjecture, Milne–Ramachandran formulated and proved similar conjectures for smooth proper schemes over finite fields. The formulation of these conjectures already relied on other unproven conjectures. In this talk, I will discuss an unconditional formulation and proof of these conjectures.

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Some results about number fields with Polya groups equal to ideal class groups

Speaker: 
Abbas Maarefparvar
Date: 
Wed, Feb 12, 2025
Location: 
Online
Zoom
Abstract: 

The Polya group of a number field K is a specific subgroup of the ideal class group Cl(K) of K, generated by all classes of Ostrowski ideals of K. In this talk, I will discuss the equality Po(K)=Cl(K) in two directions. First, we will see this equality happens for infinitely many "non-Galois fields'' K. Accordingly, I prove two conjectures presented by Chabert and Halberstadt concerning the Polya groups of some families of non-Galois fields. Then, I present some "finiteness theorems" for the equality Po(K)=Cl(K) for some families of "Galois" fields K obtained in joint work with Amir Akbary (University of Lethbridge).

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Moments of symmetric square L-functions

Speaker: 
Dmitry Frolenkov
Date: 
Tue, Feb 4, 2025
Location: 
Online
Zoom
Abstract: 

I am going to discuss various results on moments of symmetric square L-functions and some of their applications. I will mainly focus on a recent result of R. Khan and M. Young and our improvement of it. Khan and Young proved a mean Lindelöf estimate for the second moment of Maass form symmetric-square L-functions $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the short interval of length $G >> |t_j|^{(1 + \epsilon)/t^{(2/3)}}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ as long as $|t_j|^{(6/7 + \delta)} << t < (2 - \delta)|t_j|$. We obtain a mean Lindelöf estimate for the same moment in shorter intervals, namely for $G >> |t_j|^{(1 + \epsilon)/t}$. As a corollary, we prove a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the interval $|t_j|^{(2/3 + \delta)} << t << |t_j|^{(6/7 - \delta)}$. This is joint work with Olga Balkanova.

Class: 

Classification of some Galois fields with a fixed Polya index

Speaker: 
Abbas Maarefparvar
Date: 
Thu, Jan 30, 2025
Location: 
PIMS, University of Calgary
Conference: 
UCalgary Algebra and Number Theory Seminar
Abstract: 

The Polya group P o ( K ) of a Galois number field K coincides with the subgroup of the ideal class group C l ( K ) of K consisting of all strongly ambiguous ideal classes. We prove that there are only finitely many imaginary abelian number fields K whose "Polya index" [ C l ( K ) : P o ( K ) ] is a fixed integer. Accordingly, under GRH, we completely classify all imaginary quadratic fields with the Polya indices 1 and 2. Also, we unconditionally classify all imaginary biquadratic and imaginary tri-quadratic fields with the Polya index 1. In another direction, we classify all real quadratic fields K of extended R-D type (with possibly only one more field K ) for which P o ( K ) = C l ( K ) . Our result generalizes Kazuhiro's classification of all real quadratic fields of narrow R-D type whose narrow genus numbers are equal to their narrow class numbers.

This is a joint work with Amir Akbary (University of Lethbridge).

Class: 

Convolution sums from Trace Formulae

Speaker: 
Kim Klinger-Logan
Date: 
Tue, Jan 28, 2025
Location: 
Online
Zoom
Abstract: 

Previously we found certain convolution sums of divisor functions arising from physics yield Fourier coefficients of modular forms. In this talk we will discuss the limitations of the current proof of these formulas. We will also explore the connection with the Petersson and Kuznetsov Trace Formulae and the possibility of extending these formulas to other cases. The work mentioned in this talk is in collaboration with Ksenia Fedosova, Stephen D. Miller, Danylo Radchenko, and Don Zagier.

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Non-vanishing for cubic Hecke L-functions

Speaker: 
Alexandre de Faveri
Date: 
Tue, Jan 14, 2025
Location: 
Online
Zoom
Abstract: 

I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo square-free Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment. No such asymptotic formula was previously known for a cubic family (even over function fields).

Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the family of quadratic characters and the family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

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