Past Events

Past SFU Number Theory Talks

Speaker: Nils Bruin, SFU

Title: Invisible Sha[4]

Date:  Thu 14 Nov 2013

Mazur observed that for a lot of elliptic curves E with non-trivial elements in Sha(E/Q)[n], one can find another elliptic curve E' that is n-congruent to E and for which the corresponding element in $H^1(\mathbb{Q},E'[n])$ lies in the image of the Mordell-Weil group of E'. Such an element in Sha is said to be made visible by E'.

It was since proved that for n=2,3, one can always find such an E', both when the n-congruence preserves Weil-pairing and when it inverts it. For given E and n=4, the question boils down to deciding if a certain K3 surface has a rational point. In joint work with Tom Fisher, we have been able to finally find equations for these K3 surfaces, which allows us to determine visibility computationally in specific cases.

Speaker: Imin Chen, SFU

Title: Darmon's program for x^p + y^p = z^r and first case solutions

Date:  Thu 7 Nov 2013

Darmon has developed a program to resolve the generalized Fermat equation xp + yp = zr using Galois representations and abelian varieties of GL2 type over a totally real field. I will survey some parts of his program and point out the key difficulties which remain. Recently, numerous irreducibility criteria for the mod p representations attached to elliptic curves over totally real fields have been developed (David, Billerey, Freitas-Dieulefait, Freitas-Siksek). These are based on a technique which first appeared in Serre's 1972 Inventiones paper. I will explain how this method can be adapted to Darmon's Frey abelian varieties of GL2 type over a totally real field and thereby show that the above equation does not have any non-trivial first case solutions for p large enough compared to r a regular prime ≥ 5.

Speaker:  Colin Weir, SFU

Title:  Counting dihedral function fields

Date:  Thu 24 Oct 2013

In the early 70's Davenport and Heilbronn derived the leading term in the asymptotic formula for the number of cubic number fields with bounded discriminant. However, as algorithmic data became available, a large "gap" became evident between the actual number of cubic number fields of small discriminant and the asymptotic prediction. We will discuss this and the analogous situation in the function field setting. We will present methods for constructing and tabulating dihedral function fields (which includes non-Galois cubics) and prove the existence of a similar "gap" for cubic function fields of small discriminant and the leading term of the corresponding asymptotic.

Speaker:  Michael Coons, University of Newcastle, Australia

Title:  Mahler's Method, digital expansions, and algebraic numbers (or not)

Date:  Thu 26 Sep 2013, 3:30pm

In this talk, we survey past, present, and possible future results concerning the arithmetic nature of low complexity sequences. For example, what properties can be exhibited by numbers whose base expansion can be determined by a finite automaton? In the current context, this line of questioning was unknowingly initiated by Mahler, and later championed by Loxton and van der Poorten following the work of Cobham and Mendes France. In addition to describing some historical work, this talk will describe some of the the current advancements and generalisations concerning Mahler's method.

Series: PIMS Colloquium

Speaker:  Frits Beukers, Utrecht University

Title:  What are hypergeometric functions?

Date:  Thursday, May 2, 2013

Hypergeometric functions occur in many shapes and flavours
throughout mathematics and mathematical physics. The first such
functions were introduced by Euler and studied in depth by Gauss.
Since the end of the 19th the concept of hypergeometric functions was
extended in many directions, thus creating a veritable zoo of
different functions both inone variable and several variables.
By the end of the 1980's Gel'fand, Kapranov and Zelevinsky introduced
the concept of A-hypergeometric functions,which created a remarkable
amount of order through combinatorial ideas. In this lecture we give a
first introduction to hypergeometric functions and explain the idea of
A-hypergeometric functions.

Series: PIMS Colloquium

Speaker:  Tom Archibald, SFU

Title:  The hypergeometric series and the hypergeometric equation: highlights of
their roles in classical mathematics

Date:  Thursday, May 2, 2013

Things hypergeometric reach out in various directions that
may be a little surprising. In this talk we will look at some
nineteenth-century developments. Beginning with some results of Gauss,
we will sample from work by E. E. Kummer (who, in providing solutions
for the hypergeometric equation, had noticed connections to Legendre's
period relations for elliptic integrals); and by L. Fuchs (who
characterized the hypergeometric equation among linear DEs of the
"Fuchsian" class). These studies are linked to work by Fuchs, Hermite
and others on modular equations, and the detailed history reveals some
surprising connections in classical mathematics.

PIMS Number Theory Seminar
Tuesday, May 7, 2013, SFU K9509
2:00 pm, Frits Beukers (Utrecht University)
Title: Analytic aspects of hypergeometric functions

The hypergeometric functions of Gauss formed the perfect testing ground for Riemann's ideas on analytic continuation of complex analytic functions. Many properties of hypergeometric functions became evident through the use of the so-called monodromy group. We shall explain these ideas and show some applications. Time permitting, we discuss possibilities to extend these ideas to the several variable setting.

3:00pm, Frits Beukers (Utrecht University)
Title: Arithmetic aspects of hypergeometric functions

Abstract: By the end of the 1980's several authors introduced the concept of hypergeometric function on a finite field.  Although this is a purely number theoretical finite sum, it shares many properties with its analytic counterpart. The special values of these functions turn out to be related to point counting on algebraic varieties over finite fields or better, traces of Frobenius operators. In this lecture we introduce these finite hypergeometric functions and describe some of their properties.


Number Theory Seminar
Anna Haensch, Wesleyan University
Thu 15 Mar 2012, 3:00pm
Title: A characterization of almost universal ternary inhomogeneous quadratic forms

A fundamental question in the study of integral quadratic forms is the representation problem which asks for an effective determination of the set of integers represented by a given quadratic form. A slightly different, but equally interesting problem, is the representation problem for inhomogeneous quadratic forms. In this talk, we will discuss a characterization of positive definite almost universal ternary inhomogeneous quadratic forms which satisfy some mild arithmetic conditions. Using these general results, we will then characterize almost universal ternary sums of polygonal numbers.


Adrian Belshaw, Capilano University
Thu 16 Feb 2012, 4:10pm
Strong normality

At a previous seminar, we proposed a "strong normality" test, to exclude numbers like Champernowne's number. Now we give a sharp version of this test. Almost all numbers are strongly normal, and every strongly normal number is normal. We use a method of Sierpinski to construct an absolutely normal number satisfying the new criterion. (This is joint work with Peter Borwein.)


Himadri Ganguli, SFU
Thu 16 Feb 2012, 3:00pm
On the correlation of completely multiplicative functions

Let f(n) be an arithmetic function and x > 0, then we define the correlation function C(f, x) = P n≤x f(n)f(n + 1)f(n + 2). In this talk we present an asymptotic formula for C(f, x) in the case when f(n) is a completely multiplicative function and |f(n)| ≤ 1 for all n ∈ N. Let λy(n) denote the truncated Liouville function which equals +1 or −1 according n has odd or even number of prime divisors p ≤ y counted with multiplicity. It follows from the main theorem that C(λy, x) = o(x) whenever y = x o(1) and speaks in favour of the Chowla conjecture that C(λ, x) = o(x) where λ is the classical liouville function.


Imin Chen, SFU
Thu 26 Jan 2012, 4:10pm
On the equation a^3 + b^{3n} = c^2

I will explain how to apply the modular method to resolve cases of this family of generalized Fermat equations. (joint with M. Bennett, S. Dahmen, S. Yazdani). 


Stephen Choi, SFU
Thu 26 Jan 2012, 3:00pm
On small fractional parts

This is joint work in progress with Alan Haynes and Jeffrey Vaaler. Let A be a finite, nonempty set of positive integers. For x in R/Z, we study Delta(A,x) := min { ||ax|| : a \in A }, where || y || = \min { |y-n| : n \in Z } is the distance from y to the nearest integer. If each element of A is odd, then it is obvious that Delta(A,1/2)=1/2. However, in this talk, we will show that for most points x in R/Z the value of Delta(A,x) is not much bigger than |A|-1/2.


Paul Mezo, Carleton University
Thu 17 Nov 2011, 4:10pm
Character identities in real twisted endoscopy

Part of the Langlands Program is to find a meaningful correspondence between representations of Galois groups and representations of reductive algebraic groups. I will attempt to motivate this through an example and then concentrate on what happens at a (real) Archimedean place of the global picture. In this context the idea of endoscopy arises in a natural fashion and suggests identities between representations of different Lie groups. These identities have been proven by Shelstad. I will sketch the theory of endoscopy under twisting by a group automorphism and describe character identities between discrete series representations.


Jeanine Van Order, Ecole Polytechnique Fédérale de Lausanne
Thu 17 Nov 2011, 3:00pm
Galois averages of Rankin-Selberg L-functions

I will first review the notion of Galois averages of Rankin-Selberg L-functions, in particular those of Rankin-Selberg L-functions of weight-two cusp forms times theta series associated to Hecke characters of imaginary quadratic fields. I will then present a conjecture about the behaviour of these averages with the conductor of the character, of which the nonvanishing theorems of Rohrlich, Vatsal and Cornut-Vatsal are special cases. Finally, I will explain a strategy of proof, at least in the setting where the class number is equal to one.


Mike Bennett, UBC
Thu 20 Oct 2011, 4:10pm
The generalized Fermat equation

We will survey results on, and techniques for, the generalized Fermat equation x^p + y^q = z^r, where p, q, and r satisfy 1/p + 1/q + 1/r < 1. This is joint work with Imin Chen, Sander Dahmen, and Soroosh Yazdani.


Nick Harland, UBC
Thu 20 Oct 2011, 3:00pm
The iterated Carmichael lambda function

The Carmichael lambda function \lambda(n) is defined to be the smallest positive integer m such that a^m \equiv 1 \pmod{n} for all (a,n)=1. \lambda_k(n) is defined to be the k th iterate of \lambda(n). We will discuss some previous known results about k=1,2 as well as sketch a proof of a normal order for n/\lambda_k(n) for all k. 


Nils Bruin, SFU
Thu 22 Sep 2011, 4:10pm
Imaginary quadratic class numbers and Sha on congruent number curves

We consider two classical number theoretic problems that may seem quite unrelated:
* What is the power of 2 dividing the class number of Q(sqrt(-n))
* Which n are congruent numbers (n called congruent if it occurs as the area of a right-angled triangle with rational length sides).
The second question is equivalent to determining whether the elliptic curve E_n: y^2=x^3-n^2*x has positive rank. This observation suggest we might want to consider:
* What is the power of 2 in the order of Sha(E_n).
If we restrict to prime values n=p, it is already known that partial answers to these questions can be related to the splitting of p in the quartic number field Q(sqrt(1+i)).
In this talk we will discuss the next step in the classification.


Paul Pollack, SFU/UBC
Thu 22 Sep 2011, 3:00pm
Messing with perfection

Let s(n) denote the sum of the proper divisors of n, so, e.g., s(4)=1+2=3. A natural number n is called *perfect* if s(n)=n and *amicable* if s(n) =/= n but s(s(n))=n. For example, 6 is a perfect number, and 220 is an amicable number. Questions about perfect and amicable numbers constitute some of the oldest unsolved problems in mathematics. I will talk about old and new theorems concerning these numbers and their generalizations. Some of this is joint work with Mits Kobayashi (Cal Poly Pomona), Florian Luca (Universidad Nacional Autónoma de México), and Carl Pomerance (Dartmouth College).


Matthew Smith, UBC
Thu 31 Mar 2011, 4:10pm
On additive combinatorics in higher degree systems

We consider a system of k diagonal polynomials of degrees 1, 2,..., k. Using methods developed by W.T. Gowers and refined by Green and Tao to obtain bounds in the 4-term case of Szemeredi's Theorem on long arithmetic progressions, we show that if a subset A of the natural numbers up to N of size d_N*N exhibits sufficiently small local polynomial bias, then it furnishes roughly the expected number of solutions to the given system. If A furnishes no non-trivial solutions to the system, then we show via an energy incrementing argument that there is a concentration in a Bohr set of pure degree k, and consequently in a long arithmetic progression. We show that this leads to a bound on the density d_N of the set A of the form d_N << exp(-c*sqrt(log log N)), where c>0 is a constant dependent at most on k.


Guillaume Maurin, UBC
Thu 31 Mar 2011, 3:00pm
Polynomial equations with constant coefficients over function fields

We will present new conjectures on polynomial equations with constant coefficients over a function field of arbitrary characteristic (joint work with Ghioca). These statements are inspired by previous conjectures from Zilber, Pink and Bombieri, Masser and Zannier. We will try to explain how known results on the latter may give some information on the former in the case of characteristic zero.


Greg Martin, UBC
Thu 3 Mar 2011, 4:10pm
Friable values of polynomials

We summarize the current meager state of knowledge concerning how often values of polynomials have only small prime factors (that is, the values are "friable" or "smooth"). We also present some evidence, in the form of a theorem conditional upon a suitably explicit hypothesis on prime values of polynomials, to support a conjectured asymptotic formula for the number of friable values of any polynomial. 


Alexander Molnar, SFU
Thu 3 Mar 2011, 3:00pm
Affine minimal rational functions

Many arithmetic geometric results have an arithmetic dynamic analogue. For instance, Siegel's theorem that an elliptic curve has only finitely many integer points is analogous to the fact that any orbit under a rational function whose second iterate has non-constant denominator has only finitely many distinct integer values.
A conjecture of Lang states that the number of integer points on a minimal Weierstrass model of an elliptic curve is uniformly bounded. In order to translate this conjecture, one needs a dynamic concept of minimality. We present one such notion, affine minimality, an algorithm to compute affine minimal forms of rational functions and some recent results pertaining to the dynamical analogue of Lang's conjecture.


Himadri Ganguli, SFU
Thu 27 Jan 2011, 4:10pm
On the equation f(g(x)) = f(x) h^m(x) for composite polynomials



Nils Bruin, SFU
Thu 27 Jan 2011, 3:00pm
Explicit descent setups

In modern language, Fermat's Descent Infini establishes that an elliptic curve has a Mordell-Weil group of rank 0. Since then, the method has been generalized to provide an upper bound on the rank of any elliptic curve and further work also allows the analysis of the Mordell-Weil group of Jacobians of many hyperelliptic curves. Reformulating work of Schaefer, we present a general framework, in principle applicable to any curve, which allows us, under certain technical conditions, to provide an upper bound on the rank of the Jacobian of any curve. In particular, we have been able to compute some rank bounds on Jacobians of smooth plane quartic curves. This is joint work with Bjorn Poonen and Michael Stoll.


Soroosh Yazdani, UBC
Thu 2 Dec 2010, 4:10pm
On level lowering and level raising of modular forms

Let f \in S_2(\Gamma_0(N)) be a modular newform of weight 2 and level N. Then given a prime ideal \lambda , in certain favourable cases, we can say if there is a modular newform g of weight 2 at level M such f \equiv g \pmod \lambda . When M|N this is a level lowering result, while when N |M this is a level raising result. In this talk I will discuss what happens when \lambda is not a prime ideal, but rather a power of a prime ideal.


Tasho Statev-Kaletha, Princeton University
Thu 2 Dec 2010, 3:00pm
Depth-zero local Langlands correspondence and endoscopic transfer

The local Langlands correspondence seeks to parameterize the smooth irreducible representations of a reductive group G over a local field F in terms of Langlands parameters, objects closely related to representations of the Galois group of F. Each parameter is supposed to correspond to a finite set of representations of G, called an L-packet. The broad principle of Langlands functoriality suggests that often such an L-packet \Pi_G transfers to an L-packet \Pi_H on an endoscopic group H. The transfer is encoded in identities between the characters of representations in \Pi_G and those in \Pi_H. Endoscopic character identities play an important role not only in representation theory, but also in number theory, via the stabilization of the Arthur-Selberg trace formula.
In this talk, we will motivate the problem that the theory of endoscopy addresses, and then formulate the precise statement of the endoscopic character identities, after recalling the necessary notions from the local Langlands correspondence. If time permits, we will then discuss their proof for the depth-zero supercuspidal L-packets recently constructed by DeBacker-Reeder. The main technical tool involved is Waldspurger's work on endoscopy for p-adic Lie algebras, which ultimately rests on the fundamental lemma.


Charles Samuels, PIMS/SFU/UBC
Thu 28 Oct 2010, 4:10pm
A family of polynomials connected to the Goldbach conjecture

We introduce a collection of polynomials G_N in Z[z] having the following property: the Nth cyclotomic polynomial divides G_N if and only if N cannot be represented as a sum of two odd primes. Numerical evidence suggests that, in fact, G_N is irreducible and has no roots on the unit circle. We proceed to discuss some basic properties of G_N, including giving asymptotic estimates on the size of their coefficients. At various stages, this work is joint with P. Borwein, K.K. Choi, and G. Martin.


Andreas Enge, Université Bordeaux 1
Thu 28 Oct 2010, 3:00pm
Algorithms for complex multiplication of elliptic curves
The theory of complex multiplication provides algorithms for obtaining elliptic curves over finite fields with a number of points known in advance, which finds applications in cryptography and primality proving. The main ingredient is the construction of Hilbert class fields of imaginary-quadratic number fields. While these are of exponential size with respect to the input, several approaches have been described that are quasi-linear in the output. I will give a self-contained overview of the algorithms and the latest record computations.


Jonas Jankauskas, Vilnius University/SFU
Thu 7 Oct 2010, 4:10pm
On the intersection of infinite geometric and arithmetic progressions


Stephen Choi, SFU
Thu 7 Oct 2010, 3:00pm
On the optimal L_4 norm for reciprocal unimodular polynomials



Nike Vatsal, UBC
Thu 8 Apr 2010, 4:10pm
Period integrals of modular forms

I will talk about work in progress on certain adelic period integrals of modular forms on SL_2 and GL_2. it turns out that the situation for SL_2 is quite different from that of GL_2 and we'll try to explain what some of the differences mean for non-vanishing of L-functions.


Leo Goldmakher, University of Toronto
Thu 8 Apr 2010, 3:00pm
Sharp bounds on odd-order character sums
A celebrated result of Halasz characterizes the multiplicative functions taking values in the complex unit disc which have a non-zero mean value; recent work of Granville and Soundararajan characterizes the Dirichlet characters which have large character sums. I'll describe how one can prove a hybrid of these two, and show how this leads to improvements over Granville and Soundararajan's bounds. In particular, on the assumption of the Generalized Riemann Hypothesis the method yields a sharp bound on cubic character sums.


Moshe Adrian, University of Maryland, College Park
Thu 18 Mar 2010, 4:10pm
A new construction of the tame local Langlands correspondence for GL(n,F), n a prime
In my thesis, I give a new construction of the tame local Langlands correspondence for GL(n,F), n a prime. The Local Langlands Correspondence for GL(n,F) has been proven recently by Henniart, Harris/Taylor. In the tame case, supercuspidal representations correspond to characters of elliptic tori, but the local Langlands correspondence is unnatural because it involves a twist by some character of the torus. Taking the cue from the theory of real groups, supercuspidal representations should instead be parameterized by characters of covers of tori. DeBacker has calculated the distribution characters of supercuspidal representations for GL(n,F), n prime, and they are written in terms of functions on elliptic tori. Over the reals, Harish-Chandra parameterized discrete series representations of real groups by describing their distribution characters restricted to compact tori. Those distribution characters are written down in terms of functions on a canonical double cover of real tori. I have succeeded in showing that if one writes down a natural analogue of Harish-Chandra's distribution character for p-adic groups, it is the character of a unique supercuspidal representation of GL(n,F), where n is prime, far away from the identity. These results pave the way for a new construction of the local Langlands correspondence for GL(n,F), n prime. In particular, there is no need to introduce any character twists.


Jennifer Johnson-Leung, University of Idaho
Thu 18 Mar 2010, 3:00pm
Siegel modular forms of degree two attached to Hilbert modular forms

This is joint work with Brooks Roberts. Let E be a real quadratic field and let P be a cuspidal, irreducible, automorphic representation of GL(2) of the adeles of E with trivial central character and infinity type (2, 2n+2). We show that there exists a Siegel paramodular newform F with weight, level, epsilon factor, Hecke eigenvalues and L-function determined explicitly by P. These invariants are tabulated for all choices of P. I will also discuss some applications of this result.


Mike Bennett, UBC
Thu 11 Feb 2010, 4:10pm
Effective S-unit equations and a conjecture of Newman
Given a positive integer $N$, an old problem of D.J. Newman is to bound the number of ways to express $N$ as
N = 2^a 3^b + 2^c + 3^d
in nonnegative integers $a, b, c$ and $d$. That this number is finite is a consequence of a result of Evertse on $S$-units equations. That it is at most 9 requires some new ideas. I will sketch a proof of this and attempt to show how such an odd question fits into a more general framework.


Moritz Minzlaff, Technische Universität Berlin
Thu 11 Feb 2010, 3:00pm
Computing zeta functions of superelliptic curves in larger characteristic

Computing zeta functions of curves over finite fields is an important problem in computer algebra with connections to cryptography and coding theory, among others. In this talk, I first want to highlight how rigid cohomology can be used to construct explicit algorithms and why their runtime is usually linear in the characteristic p. In a second part, I will restrict the problem to superelliptic curves and show how the complexity can be reduced to be linear in the squareroot of p.


Kate Stange, PIMS/SFU/UBC
Thu 12 Nov 2009, 4:10pm
Amicable pairs of primes for elliptic curves

Let E be an elliptic curve defined over Q. A pair of primes (p,q) is called an amicable pair for E if #E(F_p) = q and #E(F_q) = p. Although rare for non-CM curves, such pairs are relatively abundant in the CM case. I will explain the difference, present conjectures and experimental data for their frequency, discuss some generalisations and related questions, and spend some time on the still-mysterious j=0 case. This talk will afford an opportunity to use cubic reciprocity. This is joint work-in-progress with Joseph H. Silverman.


Matt Greenberg, University of Calgary
Thu 12 Nov 2009, 3:00pm
Shimura curves, p-adic L-functions and rational points on elliptic curves

In this talk on joint work with Shahab Shahabi, I would like to describe how algebraic parts of periods of cycles on Shimura curves are interpolated by certain p-adic L-functions. In appropriate situations, derivatives of these p-adic L-functions are related to Heegner and Stark-Heegner points on elliptic curves. This generalizes results of Bertolini-Darmon and Shahabi concerning the analogous situations for classical modular curves.


Nils Bruin, SFU
Thu 15 Oct 2009, 4:10pm
My summer/winter in Sydney

Apart from sharing some interesting touristic and climatological observations, I will report on the computational number theoretic improvements I have included in Magma in June 2009.
While explicit p-adic analytic methods for solving diophantine equations based on Chabauty's ideas have been available for around 10 years now, there has been a recent shift to concentrate computational effort on an additional phase that can combine p-adic information at several primes. Heuristically, the method commonly referred to as "Mordell-Weil sieving" should yield arbitrarily detailed information on the location of possible solutions. In practice, however, there are severe combinatorial obstructions to exploiting that information.
In joint work with Michael Stoll, we have developed good ways of avoiding the intermediate combinatorial explosion. This strategy has now been implemented in Magma and yields an almost completely automatic procedure to determine the rational points on a considerable class of algebraic curves.


Sander Dahmen, PIMS/SFU/UBC
Thu 15 Oct 2009, 3:00pm
On the residue class distribution of the number of prime divisors of an integer

The Liouville function is defined by $\lambda(n):=(-1)^{\Omega(n)}$ where $\Omega(n)$ is the number of prime divisors of the positive integer $n$ counting multiplicity. Let $m \geq 2$ be an integer and $\zeta_m$ be a primitive $m$--th root of unity. As a generalization of Liouville's function, we study the function $\lambda_m (n):=\zeta_m^{\Omega(n)}$. Using properties of this function, we will show how, for any integer $j$, properties about the set of all positive integers $n$ with $\Omega(n) \equiv j \pmod{m}$ can be obtained. In particular, we will show that this set has (natural) density $1/m$. In fact, we will also obtain much information about error terms and will illustrate how the case $m=2$ is very different from the case $m>2$. This is joint work with Michael Coons.


Johannes Nicaise, University of Leuven, Belgium
Thu 17 Sep 2009, 4:10pm
Trace formula for varieties over a discretely valued field

We prove a trace formula à la Grothendieck-Lefschetz-Verdier for varieties X over a henselian discretely valued field with algebraically closed residue field. To the variety X, one can associate a motivic Serre invariant S(X), which measures the set of rational points on X. The trace formula expresses this measure in terms of the Galois action on the ell-adic cohomology of X, if X satisfies a certain tameness condition. If X is a curve, we relate the trace formula to Saito's criterion for tame ramification of the cohomology of X. If X is an abelian variety, we show how the trace formula gives a cohomological expression for the number of components of the Néron model of X.


Charles Samuels, PIMS/SFU/UBC
Thu 17 Sep 2009, 3:00pm
Metric versions of Mahler's measure

The metric Mahler measure $M_1:A\to[1,\infty)$ is a modification of the classical Mahler measure $M$ that satisfies the triangle inequality $M_1(\alpha\beta)\leq M_1(\alpha)M_1(\beta)$. This function was first studied by Dubickas and Smyth in 2001, where they suggested a certain weakened version of Lehmer's conjecture. We establish this conjecture as well as give some applications showing that the value of $M_1$ cannot be too mysterious. We further examine a collection of other metric Mahler measures that give rise to new problems.