This is a day-long meeting on number theory hosted by the Magma computational algebra group at the University of Sydney, Australia on Thursday, May 12, 2011. The theme of the conference will be number theory, in part in gratitude for the time spent by Claus Fieker with the Magma group over the last decade. There will be 7 talks, given by various visitors and Magma personnel, and it will run from approximately 9:30am until 6pm. There will likely be a plan for the pub and dinner afterward -- the current picks are: The Lansdowne Hotel and the Spicy Sichuan Restaurant (in Glebe).

**Photos**

A small selection of photos is now available.

List of speakers:

Frank Calegari --- Northwestern University Brendan Creutz --- Magma Daniel Delbourgo --- Monash University Claus Fieker --- Magma Eric Mortenson --- The University of Queensland Frederick Vercauteren --- Katholieke Universiteit Leuven Jared Weinstein --- Boston University

Schedule.

Thursday, May 12, 2011 (Civil Engineering Building, Drawing Office, Room 361) 9:45am-10:30am F. Vercauteren: Fully Homomorphic Encryption using Principal Ideals in Number Rings 10:30am-11:15am B. Creutz: Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula 11:15am-11:45am *** Morning Break *** 11:45am-12:30pm J. Weinstein: Varieties over finite fields with many many points 12:30pm-2:30pm *** Lunch Break *** 2:30pm-3:15pm C. Fieker: Working in the multiplicative group of a number field 3:15pm-3:45pm E. Mortenson: Ramanujan, partitions, and mock theta functions. 3:45pm-4:15pm *** Afternoon Break *** 4:15pm-5:00pm D. Delbourgo: How the shape of K_1 affects the growth of Mordell-Weil ranks 5:00pm-5:45pm F. Calegari: Even Galois representations

**Talk titles and abstracts**

Frank Calegari: Even Galois representations.

What are the Galois extensions K/Q unramified away from p such that G = Gal(K/Q) admits an irreducible representation into GL_2(Fbar_p)? When K is complex, the answer is given by the theory of modular forms, and Serre's conjecture. In this talk, we discuss the case when K/Q is real, both from a theoretical and computational perspective.

Brendan Creutz: Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula.

Abstract: The Birch and Swinnerton-Dyer conjectural formula relates the leading term of the Taylor expansion of the L-function of an elliptic curve at s = 1 to a host of arithmetic invariants of the curve. Using various theoretical and computational tools one can verify this formula for specific elliptic curves. I will discuss one such computational tool, called second isogeny descents. This was recently used to complete the verification of the formula for all elliptic curves of rank 0 or 1 and conductor less than 5000.

Daniel Delbourgo: How the shape of K_1 affects the growth of Mordell-Weil ranks.

We explain how the Ritter-Weiss congruences between arithmetic elements in certain K-groups, can yield information on the growth of Mordell-Weil ranks of elliptic curves (as you climb the layers in a p-adic Lie extension). We also mention some computations using L-series that strongly support these K_1-congruences.

Claus Fieker: Working in the multiplicative group of a number field.

The multiplicative group of a number field is rather large and difficult to work in on a computer: we clearly do not have a finitely generated Z-module structure that can be used. Therefore to use the multiplicative group in application, one frequently starts by creating a finitely generated sub-group that is large enough to contain a solution but small enough to allow effective manipulation. Apart from the finite generation, a second problem comes from the (necessary) use of logarithms to linearize the structure, it implies that the linear structure is only approximated and not exactly represented. In this context there are a few important problems to solve: - given a finite number of non-zero number field elements, can we compute the Z-module structure? - given a tentative sub-group - can we enlarge it systematically? - given a particular problem, can we find an effective set of generators for the part that we are interested in? - given an element in the finitely generated group, can we find nicer representatives? Examples here are the computation of the class group, S-unit group, solution of norm equations, splitting of co-cycles in cohomology groups and p-Selmer group computations. I will indicate algorithmic solutions to some of the problems, classical solutions as well as new ones based on p-adic techniques.

Eric Mortenson: Ramanujan, partitions, and mock theta functions.

We will give a brief overview of the history of Ramanujan and give samplings of areas such as partitions, partition congruences, ranks, modular forms, and mock theta functions. For example: A partition of a positive number $n$ is a non-increasing sequence of positive integers whose sum is $n$. There are five partitions of the number four: 4, 3+1, 2+2, 2+1+1, 1,1,1,1. If we let $p(n)$ be the number of partitions of $n$, it turns out that $p(5n+4)\equiv 0 \pmod{5}$. How does one explain this? We will then discuss new results on mock theta functions and will show how they relate to old and recent results.

Fre Vercauteren: Fully Homomorphic Encryption using Principal Ideals in Number Rings.

In this talk, I will review the concept and possibilities of fully homomorphic encryption and describe one possible realization of such system based on principal ideals in number rings. The security of the proposed system is then related to well known problems in number theory such as computing generators of principal ideals and finding short vectors in lattices.

Jared Weinstein: Varieties over finite fields with many many points.

Over a finite field with $q^2$ elements, the projective curve with equation $y^q-y=x^{q+1}$ is ``maximal" in the sense that it has the maximum possible number of rational points relative to its genus. (This curve is called the ``Hermitian curve", and it is sometimes used in coding theory. There are also ``minimal" curves.) For higher-dimensional varieties, there is a suitable generalization of this notion: a variety is maximal when it has the maximum number of rational points allowed by the Weil conjectures (relative to its topology). We will discuss a new recipe for constructing interesting higher-dimensional varieties which we conjecture to be maximal. If our conjecture is true, these varieties constitute evidence for Tate's conjecture as well.

**Venue:**
The talks will be held in the Drawing Office of the Civil Engineering
Building (Room 361), which is sometimes called the Civil and Mining
Engineering Building. This is in the lower right corner of the map, and
is almost the first point you reach when coming from Redfern Station.

For more information: send email to Mark Watkins.

If coming, it would be nice to let us know so that we have some idea of the number of people to expect.