Web Page of the Ottawa-Carleton Number Theory Seminar

Year 2018-2019

Abstract: A singular point on a plane conic defined over Q is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola y=x^2. In this talk, we provide a measure of transcendence for singular points on conics defined over Q which, in these two cases, improves on the measure obtained by Adamczewski et Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.

Title: On Waring's problem in short intervals: the case of four fourth powers

Abstract: Consider the set S of natural numbers that can be written as a sum of four 4th powers. How large is the spacing between two elements of this set? It can be arbitrarily large, as a consequence recent work by the author. In this talk we show, on the other hand, how do the Circle Method of G.H.Hardy and J.E.Littlewood and a strategy of S.Daniel provide nontrivial upper bounds on these gaps. This work is motivated by an arithmetic study of the values of a biquadratic theta function.

Title: Revisiting the classical Kronecker limit formula after the work of R. Borcherds

Abstract: : In this talk we will revisit the so-called first Kronecker limit formula under the lens provided by some of the work of Richard Borcherds who discovered around the mid 90s a refinement to the so-called theta correspondence for special dual pairs of reductive groups. In our setting, the dual pair is simply (SL_2(R), SO(2,1)) and the integration kernel of this correspondence is played by a Siegel theta function which was introduced in 1948 in a seminal paper of Carl Ludwig Siegel in its study of indefinite quadratic forms.

Title: Diophantine exponents of approximation to a real number and to its square

Abstract: : For each real number ξ we may define four diophantine exponents which measure how "well" ξ and its square can be approximated by rational numbers or linear forms with integers coefficients. However, we do not know many families of numbers for which we can compute the associated exponents and determine the sequence of best rational approximations. First, we will present well-known examples: some of Roy's extremal numbers, Bugeaud-Laurent's Sturmian fractions and Roy's Fibonacci type numbers. Then we will give a construction which generalizes the previous constructions. The numbers so obtained are said to be of Sturmian type.

Abstract: : A Fourier-Ramanujan series is one of the form S(n)=∑_q T(q) c_q(n), where q and n run over the positive integers, and c_q(n) is a certain exponential sum called Ramanujan sum. Ramanujan (1918) found that with T(q)=1/q the resulting series S(n) is identically zero. Hardy (1921) observed that the same happens for T(q)=1/φ(q). Both formulas are equivalent to the Prime Number Theorem! We will show how to produce infinitely more examples. This is joint work with Giovanni Coppola.

Abstract: (joint with Florent Jouve) In a 1853 letter, Chebyshev noted that there seems to be more primes of the form 4n+3 than of the form 4n+1. Many generalizations of this phenomenon have been studied. In this talk we will discuss Chebyshev's bias in the context of the Chebotarev density theorem.
In this second talk on the topic, we will begin by reviewing the results presented in the first talk, and give more proof ideas. We will also focus on the generic case of S_n extensions, in which the question is strongly linked with the representation theory of this group and the ramification data of the extensions. We will see in detail how to take advantage of the rich representation theory of the symmetric group as well as bounds on characters due to Roichman, Féray, Sniady, Larsen and Shalev.

Abstract: I will begin by giving an introduction to the theory of vector-valued modular forms. I will then describe my work on the arithmetic of vector-valued modular forms with respect to a representation of Γ₀(2). The collection of vector-valued modular forms form a graded module over the graded ring of modular forms. I will explain how understanding the structure of this module allows one to show that the component functions of vector-valued modular forms satisfy an ordinary differential equation whose coefficients are modular forms. In certain cases, we can use a Hauptmodul to transform such a differential equation into a Fuchsian differential equation on the projective line minus three points. We then use the Gaussian hypergeometric series to explicitly solve this differential equation. Finally, we make use of these ideas together with some algebraic number theory to study the prime numbers that divide the denominators of the Fourier coefficients of the component functions of vector-valued modular forms.

Title: Classifying certain elliptic curves with rational 2 torsion in terms of the existence of integer points

Abstract: We continue work of ours and of Michael Bennett concerning the existence of integer points on curves with rational 2 torsion, and relate the existence to various problems in both Analytic Number Theory and Diophantine Analysis. Many open problems will be discussed along with a status report using recent theorems that follow from the work of Andrew Wiles and others on the applications of Galois representations.