Number Theory Seminar
Queen's University, Kingston, ON
Organizer: Vaidehee Thatte
Organizer: Vaidehee Thatte
Title: Fluctuations in the distribution of Hecke eigenvalues
Speaker: Neha Prabhu
Date: September 13, 2017
Abstract: A famous conjecture of Sato and Tate (now a celebrated theorem of Taylor et al) predicts that the normalised p-th Fourier coefficients of a non-CM Hecke eigenform follow the semicircle distribution as we vary the primes p. In this talk, we address a situation in which we study the fluctuations in the distribution of these Fourier coefficients about the Sato-Tate measure. This is done by averaging over families of Hecke-eigenforms of weight k. By imposing suitable conditions on the growth of the weight k, these fluctuations are shown to asymptotically follow a Gaussian distribution. This is joint work with Kaneenika Sinha.
Title: A proof of Erdos-Kac Theorem due to Granville and Soundararajan.
Speaker: Arpita Kar
Date: September 20, 2017
Abstract: Erdos-Kac theorem is one of the most important theorems of probabilistic number theory. We will look at this theorem from a historical viewpoint and outline a proof due to Andrew Granville and K. Soundararajan.
Title: A lower bound for the two-variable Artin conjecture
Speaker: François Séguin
Date: September 27, 2017
Abstract: In 1927, Artin conjectured that any integer other than -1 or a perfect square is a primitive root mod p for infinitely many p. In a 2000 paper, Moree and Stevenhagen adapted Hooley's conditional proof of Artin's conjecture to show a similar result on a two-variable analogue. During this talk, I will present three different proofs of a lower bound for this problem. This is joint work with Prof. Ram Murty.
Title: The Ubiquity of the Cauchy-Schwarz inequality
Speaker: M. Ram Murty
Date: October 4, 2017
Abstract: We will discuss the classical Cauchy-Schwarz inequality and its role in sieve theory and other parts of pure mathematics. Then, we will discuss the Heisenberg uncertainty principle in quantum mechanics as an immediate consequence of the Cauchy-Schwarz inequality. If time permits, we then move into the realm of statistics and highlight how the celebrated Cramer-Rao inequality is also a consequence of the same classical inequality of Cauchy and Schwarz.
Title: Ramification theory for degree p extensions of arbitrary valuation rings in mixed characteristic (0,p) - I
Speaker: Vaidehee Thatte
Date: October 11, 2017
Abstract: In classical ramification theory, we consider extensions of complete discrete valuation rings with perfect residue fields. We would like to study arbitrary valuation rings with possibly imperfect residue fields and possibly non-discrete valuations of rank \geq 1, since many interesting complications arise for such rings. In particular, defect may occur (i.e. we can have a non-trivial extension, such that there is no extension of the residue field or the value group). In this talk, we will discuss generalization and refinement of results in ramification theory when the residue characteristic p is positive, with focus on mixed characteristic (0,p). [Note: We may need more than one session to review the results in equal characteristic (Artin-Schreier extensions) and present the analogues in the mixed characteristic case. Hence the "- I" in the title!]
Title 1: On a conjecture of Schmid and Zhuang
Speaker 1: Ravindranathan Thangadurai (visiting)
Date: October 18, 2017
Abstract 1: Let $G$ be a finite abelian additive group with exponent of $G$ as $\exp(G)$. A fundamental constant $D(G)$ attached to $G$ (is called `Davenport Constant) is defined to be the least positive integer $t$ such that any sequence $S$ over $G$ of length $t$ has a subsequence whose sum is the zero element in $G$ (such a subsequence is called zero-sum subsequence). This constant naturally arises from a question in Algebraic Number Theory. By the structure theorem of finite abelian groups $G$, we know that $$ G\cong C_{n_1}+\cdots +C_{n_r}$$ where $n_i >1$ integers and $n_i$ divides $n_{i+1}$. Then it is easy to see that $$ D^*(G) := 1+\sum_{i=1}^r(n_i-1) \leq D(G) \leq |G| $$ which implies $D(G) = |G|$ if and only if $G =C_n$. In 1969, Olson proved that $D(G) = D^*(G)$ for all finite abelian $p$-groups. There is another related constant $\eta(G)$ which is defined as the least positive integer $t$ such that any given sequence $S$ of length $t$ has a zero-sum subsequence of length $\leq \exp(G)$. The value of this constant is known for $G$ with rank $\leq 2$ and in general it is unknown. In 2010, Schmid and Zhuang conjectured that when $G$ is a finite abelian $p$-group having $D(G) \leq 2\exp(G)-1$, then $$ \eta(G) = 2D(G) - \exp(G).$$ In this talk, we prove this conjecture for most of the finite abelian $p$-group satisfying the conditions.
Title 2: Value distribution of L-functions
Speaker 2: Anup Dixit (visiting)
Date: October 18, 2017
Abstract 2: Nevanlinna theory establishes that if two meromorphic functions share five values, then they must be the same. Replacing meromorphic functions with L-functions in the Selberg class, M.R. Murty and V.K. Murty proved that any two L-functions sharing the 0-value counting multiplicity must be the same. Moreover, J. Steuding showed that two L-functions in the Selberg class sharing two values ignoring multiplicity must be the same. In this talk, we show similar results in a larger family of L-functions, namely the Lindelof class. We also show a new uniqueness result in the Lindelof class and therefore the Selberg class.
Title: Galois Representations of CM Elliptic Curves
Speaker: Richard Leyland
Date: October 25, 2017
Abstract: We introduce the notion of the mod-N Galois representation $\rho_{E/F,N}$ attached to an elliptic curve $E/F$. Motivated by conjectures of Frey and Mazur, we aim to determine all elliptic curves $E’/F$ such that $\rho_{E’/F,N}\cong \rho_{E/F,N}$ in the case where both $E$ and $E’$ have complex multiplication. In this talk, we introduce the problems, terminology currently being worked on and present a small result in the case where $E$ and $E’$ have complex multiplication by different imaginary quadratic fields.
Title: On a conjecture of Erdos
Speaker: Siddhi Pathak
Date: November 1, 2017
Abstract: In a written correspondence with A. Livingston in the 1960s, Erdos conjectured that for an arithmetic function f, periodic with period q satisfying: (i) f(n) is 1 or -1 if q does not divide n and (ii) f(n) = 0 if q divides n, the series \sum_{n=1}^{\infty} f(n)/n is not zero (and hence, evaluates to a transcendental number) whenever it converges. In 2007, this conjecture was proved by M. Ram Murty and N. Saradha for q congruent to 3 modulo 4 and is still open when q is congruent to 1 modulo 4. In this talk, we present some new developments toward this conjecture.
Title: Smooth Sums over Smooth k-Free Integers
Speaker: Francesco Cellarosi
Date: November 8, 2017
Abstract: We provide an asymptotic estimate for certain sums over k-free integers with small prime factors. These sums depend upon a complex parameter $\alpha$ and involve a smooth cut-off $f$. They are a variation of several classical number-theoretical sums. One term in the asymptotics is an integral operator whose kernel is the $\alpha$-convolution of the Dickman-de Bruijn distribution, and the other term is explicitly estimated. The trade-off between the value of $\alpha$ and the regularity of $f$ is discussed.
Title: Base change and the Langlands reciprocity conjecture
Speaker: Peng-Jie Wong
Date: November 15, 2017
Abstract: In light of Artin reciprocity, Langlands enunciated a reciprocity conjecture asserting that any complex Galois representation is automorphic. Around 1980, Langlands established certain cyclic base change and proved his reciprocity conjecture for 2-dimensional Galois representations with projective image isomorphic to $A_4$. In this talk, we will try to give a motivation of the Langlands reciprocity conjecture and discuss its relation to base change. If time permits, we will explain how the conjectural base change leads to Langlands reciprocity for all 3-dimensional Galois representations with solvable image. (This is a joint work with M. Ram Murty and V. Kumar Murty)
Title: Cyclotomic units, values of the Riemann zeta function and p-adic measures
Speaker: Jung-Jo Lee
Date: November 22, 2017
Abstract: I would explain the connection between the cyclotomic units and the values of complex Riemann zeta function via higher logarithmic derivative maps. We can use Mahler transform to construct a p-adic zeta function. Note: I consider a series of talks, each "hopefully" self-contained.
Talk 2 : Euler system of cyclotomic units and Iwasawa's main conjecture
Talk 3 : Euler system of Heegner points and Birch and Swinnerton-Dyer conjecture
Four Speakers!
Date: November 29, 2017
Title 1: Hilbert's tenth problem over number fields
Speaker1 : M. Ram Murty
Abstract1 : Hilbert's tenth problem for rings of integers of number fields remains open in general, although a conditional negative solution was obtained by Mazur and Rubin assuming some unproved conjectures about the Shafarevich-Tate groups of elliptic curves. In this talk, we highlight how the non-vanishing of certain L-functions is related to this problem. In particular, we show that Hilbert's tenth problem for rings of integers of number fields is unsolvable assuming the automorphy of L-functions attached to elliptic curves and the rank part of the Birch and Swinnerton-Dyer conjecture. This is joint work with Hector Pasten.
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Title 2: A lower bound for the two-variable Artin conjecture
Speaker 2: François Séguin
Abstract 2: In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group Z/pZ× for infinitely many p. In a 2000 article, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a 1967 proof by Hooley for the original conjecture. During this talk, we will prove an unconditional lower bound for this two-variable problem. This is joint work with Ram Murty and Cameron Stewart.
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Title 3: On a Conjecture of Bateman about $r_5(n)$.
Speaker 3: Arpita Kar
Abstract 3: Let $r_5(n)$ be the number of ways of writing $n$ as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum $$\sum_{|j| \leq \sqrt{n}}\sigma(n-j^2)$$ where $\sigma(n)$ is the sum of positive divisors of $n$. We give a proof of Bateman's conjecture in the case $n$ is square-free and congruent to $1$ (mod $4$). This is joint work with Prof. Ram Murty.
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Title 4: Derivatives of L-series and generalized Stieltjes constants
Speaker 4: Siddhi Pathak
Abstract 4: Generalized Stieltjes constants occur as coefficients of (s-1)^k in the Laurent series expansion of certain Dirichlet series around s=1. The connection between these generalized Stieltjes constants and derivatives of L(s,f) for periodic arithmetical functions f, at s=1 is known. We utilize this link to throw light on the arithmetic nature of L'(1,f) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p-7)/2 of the generalized Stieltjes constants, { gamma_1(a,p) : 1 \leq a < p }, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath.
Title: The p-adic zeta function and Iwasawa's main conjecture
Speaker: Jung-Jo Lee
Date: January 17, 2018
Abstract: I would explain the role of the p-adic zeta function in describing the structure of certain Iwasawa module.
Title: Heights of elliptic curves and the elliptic analogue of the two-variable Artin conjecture.
Speaker: François Séguin
Date: January 24, 2018
Abstract: Similar to the way Lang and Trotter adapted Artin's primitive root conjecture in the case of elliptic curves, we consider this natural adaptation for the two-variable Artin Conjecture. In light of our recent results for the two-variable setting, we present similar, unconditional lower bounds for this elliptic analogue.
Title: Defect Extensions - I
Speaker: Vaidehee Thatte
Date: January 31, 2018
Abstract: Let $K$ be a valued field of characteristic $p > 0$ with henselian valuation ring $A$. Let $L$ be a non-trivial Artin-Schreier extension of $K$ with $B$ as the integral closure of $A$ in $L$. In the classical theory of complete discrete valuation rings, $B$ is generated as an $A$-algebra by a single element. This in particular, is not true in the defect case. We will discuss a result that allows us to write $B$ as a "filtered union over $A$", when there is defect. Similar results can be obtained in the mixed characteristic case.
Title: Different approaches to the composition law for binary quadratic forms
Speaker: Francois Seguin
Date: May 3, 2017
Abstract: During this short talk, we will review the basic notions of binary quadratic forms, as well as introduce several ways to realize the composition law. In particular, we will try to focus on Bhargava's higher composition law in this specific setting.
Title: A geometric approach to the class number problem for function fields.
Speaker: Francois Seguin
Date: May 10, 2017
Abstract: Gauss formulated a famous conjecture about the occurrence of quadratic number fields of class number one. In this talk, we will present an analogue of this conjecture in the function fields setting. We will then present a potential geometric approach to this problem, as well as the possible obstructions that arise.
Title: Dirichlet series and hyperelliptic curves
Speaker: Jung-Jo Lee
Date: May 17, 2017
Abstract: We study the number of rational points on hyperelliptic curves by studying the convergence properties of its associated Dirichlet series. This is a joint work with professor Ram Murty.
Title: A Short History of Class Field Theory
Speaker: M. Ram Murty
Date: May 24, 2017
Abstract: This is a documentary that I prepared for the ATM School on Class Field Theory, being held in India this month. It is about 54 minutes long and surves developments from Fermat to Langlands.
Title: Shimura curves and an abc bound
Speaker: Hector Pasten (visiting)
Date: May 31, 2017
Abstract: I will sketch a proof of an upper bound for the number of divisors of abc which is polynomial on rad(abc), for all triples of coprime positive integers a,b,c with a+b=c. This proves a partial result for the abc conjecture.
Title: A probable path to higher Mahler measure of cyclotomic polynomial
Speaker: Arunabha Biswas
Date: June 7, 2017
Abstract: Higher Mahler measure was introduced in 2008/2009. But $z+c$ (for any complex constant $c$) and $z+1/z+r$ (for any real constant $r$) are the only two polynomials whose k-higher Mahler measures (for any positive integer $k$) are known. In this talk we consider the polynomial $P_n(z):=(z^n-1)/(z-1)$ and try to show a few results that might be useful in finding k-higher Mahler measure of $P_n$ and possibly cyclotomic polynomials. (This is a joint work with Prof. Ram Murty).
Title: Towards positive density of quartic polynomials yielding extensions of class number one.
Speaker: Francois Seguin
Date: June 14, 2017
Abstract: Given a polynomial over a finite field, we can define a quadratic extension of the function field $F_q(t)$, and ask how often those have class number one. During this talk, we will focus on the case where the polynomials are of degree 4, present some previous results about this question, and outline an unpublished proof by R. Murty and R. Gupta suggesting positive density of such polynomials.
Title: A few comments on higher Mahler measure of $P_n(z):=(z^n-1)/(z-1)$.
Speaker: Arunabha Biswas
Date: June 21, 2017
Abstract: I shall discuss a few known results about higher Mahler measure of $P_n$ and motivation behind a conjecture about the generating function of $P_n$.
Title: No seminar
Date: June 28, 2017
Title: The two variables Artin's conjecture
Speaker: François Séguin
Date: July 5, 2017
Abstract: In 1967, Hooley published a conditional proof of Artin's conjecture on primitive roots. The argument was then used to prove conditionally several other generalized cases. We will present the two variables case, explaining how Hooley's argument was adapted in this case.
Title: Solving polynomial equations in primes and in almost primes
Speaker: Shuntaro Yamagshi
Date: July 12, 2017
Abstract: Given polynomial equations we might be interested in finding solutions in primes or in almost primes. For example, the twin prime conjecture is regarding finding infinitely many prime solutions to the equation $x_1 - x_2 = 2$. I will discuss some recent progress made on solving higher degree polynomial equations in primes and also in almost primes.
Title 1: Generalizations of Liouville and Roth’s theorems to higher dimensions
Speaker 1: Mike Roth
Date: July 19, 2017
Abstract 1: In a previous talk in the number theory seminar I discussed approximation on the real line. In this talk we will discuss a generalization of those results to algebraic varieties of arbitrary dimension.
Title 2: Parametrizing binary quartic forms with small Galois group
Speaker 2: Stan Xiao (Oxford)
Date: July 19, 2017
Abstract 2: We give two different parametrization of binary quartic forms with a rational automorphism, whose irreducible elements are quartic forms whose Galois group does not contain an element of order 3. We shall also define a height which allows us to count these quartic forms with some additional fixed data. This is joint work with Cindy Tsang.
Title: No seminar
Date: July 26, 2017
Title: Unnormalized differences and fractional parts of zeros of the derivative of the Riemann $\xi$ function
Speaker: Arindam Roy (visiting)
Date: August 2, 2017
Abstract: For the completed Riemann zeta function $\xi(s)$, it is known that the Riemann hypothesis for $\xi(s)$ implies the Riemann hypothesis for $\xi^{(m)}(s)$, where $m$ is any positive integer. Local spacing distribution of zeros of the derivative of the Riemann $\xi$ function has connection with Landau-Siegel zeros. Motivated by this connection, we study the distribution of the unnormalized differences between imaginary parts of zeros of the derivative of the Riemann $\xi$ function. We also investigate the distribution of the fractional parts of $\a\g$, where $\a$ is a fixed non-zero real number and $\g$ runs over the imaginary parts of zeros of derivative of the Riemann $\xi$ function.
Title: Wieferich's Criterion on Elliptic Curves
Speaker: François Séguin
Date: August 9, 2017
Abstract: In 1909, Wieferich proved that if a prime satisfies a simple congruence relation, it also satisfies the first case of Fermat's Last Theorem. Even with Fermat's Last Theorem being proven, it is still unknown whether infinitely many primes satisfy this congruence relation. Since then, the problem was generalized to other types of groups. During this talk, we will focus on some results of Silverman when considering this problem over elliptic curves.
Title: A problem of Chowla and generalizations
Speaker: Siddhi Pathak
Date: August 16, 2017
Abstract: In the early 1960s, S. Chowla raised a question about the non-vanishing of a Dirichlet series with periodic coefficients at s=1. In this talk, we will discuss this problem and some generalizations.
Title: Hilbert's tenth problem over number fields
Speaker: M. Ram Murty
Date*: August 24*, 2017
Abstract: Hilbert's tenth problem asking for a universal algorithm to determine whether any Diophantine equation has an integer solution has been shown to have a negative solution through the combined work of Matiyasevich, Davis, Putnam and Robinson. The same negative resolution is expected to hold over all algebraic number fields but this is still open, in general. We will discuss some recent joint work with Hector Pasten on this question and indicate how it relates to the non-vanishing of certain L-functions.
* Moved to a diffrent day and time slot 2:30 - 4 pm due to a conference.
Title: On the transcendental nature of certain generalized Euler-Lehmer constants
Speaker: Siddhi Pathak
Date: January 13, 2017
Abstract: In this talk, we present a conditional result about the arithmetic nature of generalized Euler Lehmer constants, \gamma_1(a,p) for a between 1 and p -1, when p is an odd prime greater than 5. This is joint work with Prof. Ram Murty.
Title: Euclidean Rings and Wieferich Primes
Speaker: M. Ram Murty
Date: January 20, 2017
Abstract: It is conjectured that if the ring of integers of a real quadratic field is a PID, then it is in fact a Euclidean domain. The conjecture is known under the assumption of the generalized Riemann hypothesis (GRH). We replace the GRH with another hypothesis about Wieferich primes, namely that the number of primes p < x such that 2^{p-1} = 1(mod p^2) is o(x/log^2 x). We show that this implies the conjecture. This is joint work with K. Srinivas and M. Subramani.
Title: Some Examples of Artin-Schreier Extensions of Valuation Fields ("No Theory - II")
Speaker: Vaidehee Thatte
Date: January 27, 2017
Abstract: We will discuss a few examples of Artin-Schreier extensions of valuation fields K of prime characteristic p and explicitly compute some invariants of ramification theory in each case. This is a continuation of the first "No Theory" talk.
Note/Recall: In the last talk, we discussed the basic structure of such extensions and properties of valuations involved.
Title: Class Field Theory and Complex Multiplication
Speaker: Richard Leyland
Date: February 3, 2017
Abstract: An elliptic curve is said to have complex multiplication if its endomorphism rings is strictly larger than the ring of integers. Class field theory classifies abelian extensions of a field K unramified outside a given finite list of primes. In this talk, we shall see that the theory of complex multiplication gives these extensions explicity for the case where K is an imaginary quadratic field.
Title: Twin primes and the parity problem
Speaker: Akshaa Vatwani (Waterloo)
Date: February 10, 2017
Abstract: We relate the twin prime problem to conjectures of Chowla type regarding the Mobius function. This is joint work with Professor Ram Murty.
Title: Erdős discrepancy problem and Tao's work
Speaker: Arunabha Biswas
Date: February 17, 2017
Abstract: In 1930s, Erdős speculated that every sequence of {-1,+1} assigned to natural numbers must have large "imbalances" among multiples of some integer. More than eight decades later, Tao confirmed this speculation is correct. In this talk I shall tell a few stories about this magnificent problem.
No seminar (Reading Week)
Date: February 24, 2017
Title: Some Applications of Pretentiousness in the Theory of Dirichlet Characters
Speaker: Sacha Mangerel (U Toronto)
Date: March 3, 2017
Abstract: ''Pretentious'' methods in analytic number theory, as introduced by Granville and Soundararajan, are powerful tools in the analysis of mean values and correlations of multiplicative functions. In this talk, we will give two applications of these tools to problems involving Dirichlet characters. i) For a non-principal Dirichlet character $\chi$ modulo $q$, define $M(\chi)=\max_{t } |\sum_{n \leq t} \chi(n)|$. The classical P\'{o}lya-Vinogradov inequality asserts that $M(\chi) \ll \sqrt{q} \log q$ unconditionally, and on the Generalized Riemann Hypothesis (GRH), Montgomery and Vaughan showed that $M(\chi) \ll \sqrt{q} \log \log q$. We discuss a recent improvement, in joint work with Y. Lamzouri, to both of these results in the case that $\chi$ has odd order. This improvement on GRH is best possible up to a factor of $\log \log \log \log q$. One of the key ingredients in the proof of the upper bounds is a new Hal\'asz-type inequality for logarithmic mean values of completely multiplicative functions. ii) Time permitting, we will also discuss the following rigidity theorem related to binary correlations of Dirichlet characters. If $f : \mathbb{N} \rightarrow \mathbb{C}$ is a 1-bounded multiplicative function for which there is a primitive Dirichlet character $\chi$ of conductor $q$ such that $$\sum_{n \leq x} f(n)\bar{f(n+h)} = (1+o(1))\sum_{n \leq x} \chi(n)\bar{\chi(n+h)}$$ for all shifts $|h| \leq H$, then if $H \rightarrow \infty$ with $x$ then $f$ ''pretends'' to be $\chi$ in a precise sense. This is joint work with O. Klurman.
Title: AN INTRODUCTION TO THE RANKIN-SELBERG CONVOLUTION
Speaker: M. Ram Murty
Date: March 10, 2017
Abstract: We will give a gentle introduction to the theory of the Rankin-Selberg convolution L-series, derive its analytic continuation and functional equation as well as some of its special values (if time permits).
Title: On the Riemann zeta function and Brownian excursions.
Speaker: Francesco Cellarosi
Date: March 17, 2017
Abstract: Following a 2001 paper by P. Biane, J. Pitman and M. Yor, I will review some results connecting the Riemann zeta function and the Jacobi theta function to some probability laws governing sums of independent exponential random variables. These laws are related to the 1-dimensional Brownian motion and higher dimensional Bessel processes.
Title: A Liouville approximation theorem for varieties and an application to cubic surfaces
Speaker: Mike Roth
Date: March 24, 2017
Abstract: Diophantine approximation studies the problem of approximating an algebraic point by rational ones. Results in Diophantine approximation are used to deduce Diophantine results from geometric conditions. The first results on Diophantine approximation were by Dirichlet and Liouville, and concern approximation on the real line. Liouville’s result was later greatly improved by Klaus Roth. In general Roth’s theorem is much better than Liouville’s theorem, the exception being when the point being approximated is itself rational. This talk will give an extension of Liouville’s theorem to algebraic varieties, and an application to cubic surfaces.
Title: A conjecture of Bateman regarding r_5(n)
Speaker: Arpita Kar
Date: March 31, 2017
Abstract: We will discuss about a conjecture made by Paul T. Bateman for the exact formula for $\sum{|j| \leq n}\sigma(n-j^2)$ and talk about some recent developments on the conjecture. This is joint work with Prof. Ram Murty.
Title: The circle method in function fields
Speaker: Yu-Ru Liu (U Waterloo)
Date: April 7, 2017
Abstract: We will introduce the circle method in function fields, and then apply it to study Waring's problem and Vinogradov's mean value theorem for polynomials. We will address possible problems arising in a positive characteristic setting. This is joint work with T. Wooley, W. Kuo and X. Zhao.
Title: Introduction to Ramification Groups
Speaker: Vaidehee Thatte
Date: September 16, 2016
Abstract: Classical ramification theory deals with complete discrete valuation fields with perfect residue fields. We will review the construction and properties of ramification groups (lower and upper/higher) in this case. We will briefly discuss the Hasse-Arf Theorem, if time permits.
Title: Some applications of Ratner’s theory
Speaker: Francesco Cellarosi
Date: September 23, 2016
Abstract: I will give an introduction to some results by M. Ratner in homogeneous dynamics, and discuss a few applications to Diophantine approximation. Specifically, I will discuss G. Margulis’s proof of Oppenheim’s conjecture and —time permitting— the contribution of M. Einsiedler, A. Katok and E. Lindenstrauss toward Littlewood’s conjecture.
Title: Generalized Euler-Lehmer constants
Speaker: Siddhi Pathak
Date: September 30, 2016
Abstract: Generalized Euler constants arise in the Laurent expansion of the Riemann zeta function about s=1. We will define their analogues for arithmetic progressions and discuss their basic properties, connection to Dirichlet series with periodic co-efficients and derivatives of Dirichlet L-functions at s = 1, values of higher analogues of the digamma function. Time permitting, I will briefly sketch the known results about their arithmetic nature.
Title: Effective equidistribution of preimages of iterates of the Farey map
Speaker: Byron Heersink (UIUC)
Date: October 7, 2016
Abstract: Using techniques from infinite ergodic theory, Kesseb\"ohmer and Stratmann proved equidistribution results for sets of the form $F^{-n}[\alpha,\beta]$, where $[\alpha,\beta]\subseteq(0,1]$ and $F$ is the Farey map, as well as weighted subsets of the Stern-Brocot sequence. This talk presents effective versions of these results, employing certain properties of the transfer operator of the Farey map and Freud's effective version of Karamata's Tauberian theorem. We will focus primarily on a special case which establishes an effective estimate for the Lebesgue measure of the sum-level sets for continued fractions.
Title: Finite Ramanujan expansions and shifted convolution sums
Speaker: Giovanni Coppola (Visiting)
Date: October 14, 2016
Abstract: We will talk about recent joint work with Ram Murty on a particular property of shifted convolution sums, for any couple of arithmetic functions f,g: recall that the SCS, abbrev.for shifted convolution sum, of f & g is defined, in our setting, as a sum of their values over N integers with a shift, i.e. h, between their arguments. In fact a simple remark makes it possible to truncate the divisors of both f & g in terms of N & h (this amounts to cut the supports of their 'Eratosthenes transforms', say f', g', where f(n) is the sum of f'(d) over d dividing n): this, in turn, gives the f and g FINITE Ramanujan expansions ! We study these new kind of expansions (of course, a much easier approach, with respect to series and convergence) in the study of SCS of any kind of arithmetic functions, but also we give some classical applications. Also,recently we introduced a new kind of Ramanujan expansion, namely the one with respect to the shift h (i.e., we see the SCS itself as a function of h) and we are comparing, these two kind of Ramanujan expansions (the finite ones, for the single f & g,with the shift-Ramanujan expansion) giving new results, linked to the 'regularity' w.r.t. h, the shift. This shift-Ramanujan expansion has already in the literature many known heuristic formulae !
Title: Sets of multiples
Speaker: Maria Avdeeva
Date: October 21, 2016
Abstract: In this talk, we will review different properties of the so-called sets of multiples. We will cover several results on existence and bounds on different types of densities for such sets. If time permits, we will include the Heilbronn-Rohrbach and Behrend inequalities, as well as the Erd\”os-Hall-Tenenbaum Criterion in our discussion. The talk will mostly follow the initial chapters of the book ``Sets of Multiples’’ by Richard Hall.
Title: Applications of Group Theory to a Conjecture Langlands
Speaker: Peng-Jie Wong
Date: October 28, 2016
Abstract: Let $\rho$ be an irreducible continuous Galois representation of a number field $k$ with finite image $G$. The Langlands reciprocity conjecture asserts that $\rho$ is automorphic over $k$. If $\rho$ is 1-dimensional, then this conjecture follows from Artin reciprocity. For $\rho$ 2-dimensional and $G$ solvable, the reciprocity has also been established by Langlands and Tunnell. In a slightly different vein, via their theory of base change and automorphic induction, Arthur and Clozel derived the Langlands reciprocity if $G$ is nilpotent.
Title: Intersections of Humbert surfaces and binary quadratic forms
Speaker: Ernst Kani
Date: November 4, 2016
Abstract: Humbert surfaces are certain surfaces embedded in the moduli space M_2 of genus 2 curves. In this talk I will explain the connection between components of the intersection of two Humbert surfaces and classes of certain binary qudratic forms. Using the reduction theory of binary quadratic forms, this gives a method for computing these components. In addition, this leads to (new) interesting questions about binary quadratic forms.
Title: When does a quadratic form represent all positive integers?
Speaker: Arpita Kar
Date: November 11, 2016
Abstract: In 1770, Lagrange proved that every positive integer can be expressed as sum of four squares. Later, in 1916, Ramanujan wrote a classical paper that gives 53 more ways to represent positive integers. It is natural to ask when all positive integers can be represented by a quadratic form? Indeed, this has been answered by Conway and Schneeberger. In this talk, I will discuss this theorem and sketch a simple proof due to Manjul Bhargava.
Title: Artin-Schreier Extensions of Valuation Fields - Some Examples
Speaker: Vaidehee Thatte
Date: November 18, 2016
Abstract: Let K be a field of prime characteristic p. Any degree p Galois extension L of K is the splitting field of an Artin–Schreier polynomial, i.e., a polynomial of the form X^p-X-f, where f is an element of K. We will discuss a few examples of such extensions of valuation fields and explicitly compute some invariants of ramification theory in each case.
Title: Green-Tao Theorem
Speaker: Shuntaro Yamagishi
Date: November 25, 2016
Abstract: In this talk I will try to explain the proof of the celebrated Green-Tao Theorem, which asserts that primes contain arbitrarily long arithmetic progressions.
Title: Survey of class number divisibility results
Speaker: Francois Seguin
Date: December 2, 2016
Abstract: We have previously explored the problem of counting the quadratic extensions with class number one of both number fields and function fields. This time, we will try to present results about the "other" ones, i.e. extensions with class number > 1. Specifically, we will be looking at the proportion of extensions that have a class number divisible by a certain integer. This talk will not contain any new result towards this, but rather will be a brief exposition of the different known results for quadratic and higher degree extensions of number fields and function fields.
Title: Ramification theory for arbitrary valuation rings in positive characteristic
Speaker: Vaidehee Thatte
Date: July 20, 2016
Abstract: Our goal is to develop ramification theory for arbitrary valuation fields, that is compatible with the classical theory of complete discrete valuation fields with perfect residue fields. We consider fields with more general (possibly non-discrete) valuations and arbitrary (possibly imperfect) residue fields. The defect case, i.e., the case where there is no extension of either the residue field or the value group, gives rise to many interesting complications.We present some new results for Artin- Schreier extensions of valuation fields in positive characteristic. These results relate the "higher ramification ideal" of the extension with the ideal generated by the inverses of Artin-Schreier generators via the norm map. We also introduce a generalization and further refinement of Kato's refined Swan conductor in this case. Similar results are true in the mixed characteristic case.
Title: Polynomial equations of many variables in primes
Speaker: Shuntaro Yamagishi
Date: July 27, 2016
Abstract: Given polynomials with integer coefficients, finding its zeros in primes is an important topic in number theory. In this talk, I will introduce some well known results and recent progress in this area.
Title: Twin primes and Chowla's parity conjectures
Speaker: Ram Murty
Date: August 3, 2016
Abstract: We relate the twin prime problem to various generalizations of Chowla's conjectures regarding the Mobius and Liouville functions. This is joint work the Akshaa Vatwani.
Title: On the Sato-Tate conjecture
Speaker: Peng-Jie Wong (Queen's)
Date: August 10, 2016
Abstract: The famous Sato-Tate conjecture asserts that given any non-CM elliptic curve E defined over rational number field, the Frobenius angles of E are “equidistributed”. In this talk, we will discuss how mathematicians tried to attack this conjecture and sketch the proof due to Taylor and his school. If time allows, we will give an extension of the Sato-Tate theorem with the Chebotarev condition that concerns the distribution of Frobenius angles and Artin symbols.
Title: The class number problem for quadratic function fields
Speaker: Francois Seguin (Queen's)
Date: August 17, 2016
Abstract: Gauss formulated a famous conjecture about the occurrence of quadratic number fields of class number one. In this talk, we will present an analogue of this conjecture in the function fields setting, as well as sketch a potential approach to attack this conjecture.
Title: Moduli Spaces in Number Theory
Speaker: Richard Leyland (Queen's)
Date: August 24, 2016
Abstract: The idea of a moduli space is to have a geometric object that parametrizes certain classes of families over a base. In this talk, we introduce the the basics of moduli spaces, the difficulties faced when trying to construct them and give a few examples that arise in arithmetic geometry.