Forthcoming events in this series
Complex multiplication
Abstract
In this talk I will introduce some of the basic ideas linking the theory of complex multiplication for elliptic curves and class field theory. Time permitting, I'll mention Shimura and Taniyama's work on the case of abelian varieties.
Primes in short arithmetic progressions
Abstract
The Siegel-Walfisz theorem gives an asymptotic estimate for the number of primes in an arithmetic progression, provided the modulus of the progression is small in comparison with the length of the progression. Counting primes is harder when the modulus is not so small compared to the length, but estimates such as Linnik's constant and the Brun-Titchmarsh theorem give us some information. We aim to look in particular at upper bounds for the number of primes in such a progression, and improving the Brun-Titchmarsh bound.
Conics on the Fermat quintic threefold
Abstract
(Note that the talk will be in L2 and not the usual SR1)
Many interesting features of algebraic varieties are encoded in the spaces of rational curves that they contain. For instance, a smooth cubic surface in complex projective three-dimensional space contains exactly 27 lines; exploiting the configuration of these lines it is possible to find a (rational) parameterization of the points of the cubic by the points in the complex projective plane.
After a general overview, we focus on the Fermat quintic threefold X, namely the hypersurface in four-dimensional projective space with equation x^5+y^5+z^5+u^5+v^5=0. The space of lines on X is well-known. I will explain how to use a mix of algebraic geometry, number theory and computer-assisted calculations to study the space of conics on X.
This talk is based on joint work with R. Heath-Brown.
Looking at Elliptic L-functions via Modular Symbols
Abstract
We have seen that L-functions of elliptic curves of conductor N coincide exactly with L-functions of weight 2 newforms of level N from the Modularity Theorem. We will show how, using modular symbols, we can explicitly compute bases of newforms of a given level, and thus investigate L-functions of an elliptic curve of given conductor. In particular, such calculations allow us to numerically test the Birch-Swinnerton-Dyer conjecture.
Galois representations III: Eichler-Shimura theory
Abstract
In the first half of the talk we explain - in very broad terms - how the objects defined in the previous meetings are linked with each other. We will motivate this 'big picture' by briefly discussing class field theory and the Artin conjecture for L-functions. In the second part we focus on a particular aspect of the theory, namely the L-function preserving construction of elliptic curves from weight 2 newforms via Eichler-Shimura theory. Assuming the Modularity theorem we obtain a proof of the Hasse-Weil conjecture.
Modularity and Galois representations
Abstract
This talk is the second in a series of an elementary introduction to the ideas unifying elliptic curves, modular forms and Galois representations. I will discuss what it means for an elliptic curve to be modular and what type of representations one associates to such objects.
Fast reduction in the de Rham cohomology groups of projective hypersurfaces
Abstract
Let $X$ be a smooth hypersurface in projective space over a field $K$ of characteristic zero and let $U$ denote the open complement. Then the elements of the algebraic de Rham cohomology group $H_{dR}^n(U/K)$ can be represented by $n$-forms of the form $Q \Omega / P^k$ for homogeneous polynomials $Q$ and integer pole orders $k$, where $\Omega$ is some fixed $n$-form. The problem of finding a unique representative is computationally intensive and typically based on the pre-computation of a Groebner basis. I will present a more direct approach based on elementary linear algebra. As presented, the method will apply to diagonal hypersurfaces, but it will clear that it also applies to families of projective hypersurfaces containing a diagonal fibre. Moreover, with minor modifications the method is applicable to larger classes of smooth projective hypersurfaces.
Intersections of two cubics and Artin's conjecture
Abstract
Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a
$p$-adic field $k$. A special case of a conjecture of Artin is that the
forms $C$ and $C'$ have a common zero over $k$. While the conjecture of
Artin is false in general, we try to argue that, in this case, it is
(almost) correct! This is still work in progress (joint with
Heath-Brown), so do not expect a full answer.
As a historical note, some cases of Artin's conjecture for certain
hypersurfaces are known. Moreover, Jahan analyzed the case of the
simultaneous vanishing of a cubic and a quadratic form. The approach
we follow is closely based on Jahan's approach, thus there might be
some overlap between his talk and this one. My talk will anyway be
self-contained, so I will repeat everything that I need that might
have already been said in Jahan's talk.
An Round-Up of the Circle Problem
Abstract
How many integer-points lie in a circle of radius $\sqrt{x}$?
A poor man's approximation might be $\pi x$, and indeed, the aim-of-the-game is to estimate
$$P(x) = \sharp\{(m, n) \in\mathbb{Z}: \;\; m^{2} + n^{2} \leq x\} -\pi x,$$
Once one gets the eye in to show that $P(x) = O(x^{1/2})$, the task is to graft an innings to reduce this bound as much as one can. Since the cricket-loving G. H. Hardy proved that $P(x) = O(x^{\alpha})$ can only possible hold when $\alpha \geq 1/4$ there is some room for improvement in the middle-order.
In this first match of the Junior Number Theory Seminar Series, I will present a summary of results on $P(x)$.