We give evidence for the Bloch-Kato conjecture for the convolution L-function of two elliptic modular forms. Let f be a newform of weight 2 and g be a newform of weight 2k, k £ 7, of level G₀(q) for an odd prime q such that they have irreducible modp Galois representations for p an odd prime different from q. Let M be the motive associated to the modp Galois representation r_f Är_g. We show that under suitable conditions on p

This is joint work with Kris Klosin.

To many arithmetic objects M (e.g., Dirichlet and Hecke characters, elliptic curves, modular forms...), one can associate a complex-analytic function L(M,s) defined on some right half-plane Re s >> 0 admitting meromorphic continuation to all of C and satisfying a function equation relating the values at s and k-s for a positive integer k. The value of L(M,s) at its central point s = k/2 conjecturally encodes arithmetic information about M (e.g., sizes of certain class groups, ranks of elliptic curves). After reviewing a few results on the vanishing of certain families of Dirichlet, Hecke, and modular L-functions at their central points, we discuss some p-adic analogues and their relationship to the classical cases.

The Liouville function is defined by l(n) : = (-1)^W(n) where W(n) is the number of prime divisors of n counting multiplicity. Let z_m : = e^2pi/m be a primitive m-th root of unity. As a generalization of Liouville's function, we study the functions l_m,k(n) : = z_m^kW(n). Using properties of these functions, we give a weak equidistribution result for W(n) among residue classes. More formally, we show that for any positive integer m, there exists an A > 0 such that for all j = 0,1,...,m-1, we have

Joint work with Sander Dahmen.

Visualizing an element of the Shafarevich-Tate group of an elliptic curve over a number field refers to representing this element in a certain way as a curve in an abelian variety. Mazur introduced this notion and proved that every element of order three can be visualized in an abelian surface (over the ground field). In this talk we explain the notion of visibility in more detail, show that the abelian surface in Mazur's result can actually be taken to be a jacobian of a genus 2 curve and give an explicit construction of this genus 2 curve.

This is joint work with Nils Bruin.

Let f(t) Î Q(t) be a rational function of degree at least 2. For a given rational number x₀, define x_n+1 = f(x_n) for each n ³ 0. If this sequence is not eventually periodic, then x_n+1-x_n has a primitive prime factor for all sufficiently large n. This result provides a new proof of the infinitude of primes for each rational function f of degree at least 2.

I will present the above result, along with some interesting refinements. I will also give a geometric description that suggests a question about dynamics in higher dimensions.

This is joint work with Andrew Granville.

I will describe some recent progress on bounding exponential sums with multiplicative coefficients. As an application of the method, on the assumption of the Generalized Riemann Hypothesis I will deduce a bound for cubic character sums which is best possible.

Lenstra generalized the idea of a Euclidean domain to the concept of a Euclidean ideal. The existence of a Euclidean ideal in a Dedekind domain implies that said domain has cyclic class group. He listed all the quadratic imaginary fields with an ideal that is Euclidean for the norm.

Joint work with Nick Ramsey shows that these are indeed all the Euclidean ideals in quadratic imaginary fields.

The multiplier of a periodic point for a holomorphic function on the Riemann sphere gives some information about the local dynamics: whether the periodic cycle attracts or repels nearby points, or acts unpredictably. I will discuss the moduli problem of parametrizing cubic polynomials with a marked point of period N and specified multiplier, a problem that turns out to have a lot more to do with algebraic geometry and number theory than it does with traditional complex dynamics.

The literature is rich with asymptotic formulae for the sum of multiplicative functions f(n) for n £ x. In contrast, little is known about multiplicative functions summed over intervals x < n £ x+y. We find asymptotic formulae for short sums of complex-valued multiplicative functions that are sufficiently "close" to 1 on primes p, and uniformly bounded on the prime powers. Some functions that fall into this category are s(n)/n and f(n)/n, where s denotes the sum of divisors function and f the Euler totient function.

There are many sequences with integral values arisen naturally from number theory. The goal in this talk is to study their probabilistic properties. For example, for a Î Z, m Î N with (a,m)=1, let l_a(m) be the order of a in (Z/mZ)^*. Let w( l_a(m) ) be the number of distinct prime divisors of l_a(m). A conjecture of Erdös and Pomerance states that if |a| > 1, then the quantity

This is a joint work with Y.-R. Liu

We study the variation of the trace of the Frobenius endomorphism associated to a cyclic trigonal curve of genus g over F_q as the curve varies in an irreducible component of the moduli space. We show that for q fixed and g increasing, the limiting distribution of the trace of Frobenius equals the sum of q+1 independent random variables taking the value 0 with probability 2/(q+2) and 1, e^2pi/3, e^4pi/3 each with probability q/( 3(q+2) ). This extends the work of Kurlberg and Rudnick who considered the same limit for hyperelliptic curves. We also show that when both g and q go to infinity, the normalized trace has a standard complex Gaussian distribution and how to generalize these results to p-fold covers of the projective line.

This is joint work with A. Bucur, C. David, and B. Feigon.

I will talk about finding upper bounds on L(1) where L(s) is an L-function. The value of an L-function at 1 has been an object of great historical interest. For instance, the value of the classical Dirichlet L-functions at 1 is linked to the class number of quadratic fields. With the conception of the Langland's program and the conjectures therein, there is now a much larger class of L-functions which may be studied.

Finding upper bounds for these L-functions at 1 presents new obstacles and yields many interesting applications. The main obstacle arises because we have no good control over the size of the coefficients of these L-functions. I will first describe some examples and applications to motivate the discussion and then sketch some of the main ideas behind a new upper bound. This work improves and generalizes previous results of Iwaniec, Molteni, and Brumley.

Given two reduced residue classes a and b (modq), let d(q;a,b) be the "probability", when x is "chosen randomly", that more primes up to x are congruent to a mod q than are congruent to b mod q (Rubinstein and Sarnak defined this quantity precisely as a logarithmic density). In joint work with Daniel Fiorilli (thanks to whom this eternal manuscript-in-preparation has finally seen the light of day), we give an asymptotic series for d(q;a,b) that can be used to calculate it to arbitrary precision. The asymptotic formula has theoretical ramifications as well: for example, it allows us to compare the relative sizes of the d(q;a,b) as a and b vary over residue classes (modq).

In recent work, we have developed the first polynomial-time algorithm to interpolate an unknown univariate rational polynomial f Î Q[x] into the sparsest shifted power basis. That is, we find the "sparsest shift" a such that f(x+a) has the fewest number of nonzero terms, and then explicitly compute the terms of f in the shifted power basis [1,(x-a),(x-a)²,¼]. Both steps in the algorithm work by computing over a series of fields Z/pZ for many small primes p. In order to guarantee that the crucial information about f is not lost by working modulo p, certain equalities must not hold in both the additive and multiplicative groups of Z/pZ. Our method for finding primes p that satisfy these conditions involves finding small primes in certain arithmetic progressions, which fortunately is a well-studied problem in number theory. Since the efficiency of the algorithm depends heavily on the size of the chosen primes, we need good bounds on their size. We examine the various approaches and results used to construct these small primes, and discuss some open problems and areas for further refinement.

This is joint work with Mark Giesbrecht.

I will discuss a conjecture and some partial results towards it on the relationship between cup products of cyclotomic units and p-adic L-values of cusp forms that satisfy congruences with Eisenstein series at primes over p.

Let q(x,y,z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of the integral solutions (x, y, z) with |x|, |y|, |z| £ B.

The analytic rank of the Jacobian J₀(N) of the modular curve X₀(N) is closely connected with the behaviour of the traces of Hecke operators acting on spaces of cusp forms of weight 2 and level N. We utilize this connection in order to find explicit upper bounds for the analytic rank of J₀(N).

A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways which may be of independent interest. The Erdös-Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this where the target interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

A pair of primes p and q are called amicable for an elliptic curve if the order of reduction modulo p is q and the order of reduction modulo q is p. Such pairs are, not surprisingly, relatively rare for most elliptic curves. On curves with complex multiplication, however, such pairs are quite frequent and have interesting properties. We will present theorems, conjectures, and experimental data.

This is work in progress, jointly with Joseph H. Silverman.

We shall discuss a natural notion of equivalence on the set of binary cubic forms F(x,y) with integer coefficients and non-zero discriminant. We shall then show that there are infinitely many inequivalent cubic binary forms F with content 1 for which the Thue equation F(x,y)=m has many solutions in integers x and y for infinitely many integers m.

An important problem in number theory is finding methods for computing invariants of number and function fields. These invariants include the system of fundamental units and the regulator of these fields. Finding efficient algorithms for computing these invariants is believed to be a difficult problem.

An effective way for computing the regulator is to perform arithmetic in a structure of ideals called the infrastructure. This infrastructure plays a paramount role in known algorithms for computing the regulator of quadratic, cubic and certain quartic number and function fields. An important ingredient in these algorithms is a process called ideal reduction.

In this talk, we will present an algorithm for reducing ideals in certain unit rank one function fields. This reduction algorithm incorporates ideas from known lattice basis reduction algorithms and the study of Minkowski geometry of numbers in a field of series.

This is joint work with Renate Scheidler at the University of Calgary.

We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Nondegenerate polynomials are popular objects in explicit number theory and algebraic geometry because of their connection with toric geometry. We determine the dimension of the space of nondegenerate curves, and we prove that there are exactly two curves of genus at most 3 that are not nondegenerate: one over F₂ and one over F₃, each with remarkable extremal properties.

We investigate the transcendental nature of the sum

This is joint work with M. Ram Murty.

Suppose we have a real quadratic number field of discriminant D. If we have a principal ideal I, it usually requires an exponential (in logD) amount of time to write out a generator of I in the conventional way. However, there exists a representation of this generator, called a compact representation, which can be written out in polynomial time. In this talk I discuss an algorithm for finding a compact representation when we are given an approximate value of the logarithm of the absolute value of a generator and an integral basis of I.

We apply the Green-Tao method to obtain the correct order of magnitude for the number of k-term arithmetic progressions in the set of integers represented as the sum of two squares, with a similar Roth-like theorem for subsets of positive relative density. The method generalizes readily to other similarly-sieved sets.