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Number Theory Talks
Spring, 2007
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Calendar
March 9 - Number Theory Seminar
10:40 - 11:30 am., PSA 206
David Roberts, University of Minnesota at Morris, Wild partitions and number theory
Abstract: For S a finite non-empty set of primes, let GNF(S) be the set of Galois number fields K such that all primes dividing its discriminant d(K) are in S. Certainly, NF(S) is infinite. Let GNF(S,Sym) be the subset of such fields such that Gal(K/Q) is a symmetric group. We combine the Krasner-Serre p-adic mass formula with a recent heuristic due to Bhargava to strongly support the conjecture that GNF(S,Sym) is finite. The talk will stress the combinatorics underlying the situation which are based on the new notion of "wild partition."
March 28 - Number Theory Seminar
12:40 - 1:30 pm., PSA 206
Florian Luca, UNAM, Morelia, Diophantine equations involving perfect powers and Fibonacci numbers
Abstract: Recently, Bugeaud, Mignotte and Siksek showed that the largest Fibonacci number F_n which is a perfect power is F_{12}=144=12^2. In this talk, I will present some recent results in the same spirit. For example, a product of two or more consecutive Fibonacci numbers is never a perfect power except for the trivial case F_1 \cdot F_2=1. Further, the Diophantine equation F_{m_1} \cdot F_{m_2} \cdots F_{m_t}=by^p, where the largest prime factor of b is bounded by a given number C and p>t is a prime, implies that \max\{m_i:i=1, \ldots,t} is bounded (in terms of C). For example, if C=541, then \max{m_i:i=1, \ldots,t} \le 44. The largest solution of the equation F_n\pm 1=y^p is F_6+1=8+1=3^2. The proofs of these results combine elementary arguments, estimates for linear forms in three logarithms of algebraic numbers and Wiles' modular approach to ternary Diophantine equations with high powers. This is joint work with Bugeaud, Mignotte, Shorey and Siksek.
April 4 - Number Theory Seminar
12:40 - 1:30 pm., PSA 206
Chad Awtrey, Degree 12 extensions of the p-adic numbers, part I
Abstract: It is well known that there are only finitely many extensions of Q_p (the p-adic numbers) of a given degree, up to isomorphism. And at the present, all extensions of Q_p through degree 11 have been completely classified by J. Jones and D. Roberts. See their interactive database for a complete description. The focus of these two talks will be on the current advances toward classifying degree 12 extensions of Q_2 and Q_3.
April 11 - Number Theory Seminar
12:40 - 1:30 pm., PSA 206
Chad Awtrey, Degree 12 extensions of the p-adic numbers, part II
April 25 - Number Theory Seminar
12:40 - 1:30 pm., PEBE 219
Keenan Kidwell, To p-Adic Norm with Love From Q - You Complete Me
Abstract: The objectives of this talk will be to define the p-adic valuation and norm on the rational numbers, to give a concise discussion of the completion of Q with respect to the metric d_p induced by the aforementioned norm, and to provide a modest overview of some of the basic (but nonetheless devastatingly intriguing) analytic and topological properties of the metric space (Q_p, d_p). If there is time, we also hope to take a brief look at polynomials over Q_p and the algebraic closure of the p-adics.
Autumn, 2006
August 23 - Number Theory and Class Field Theory Seminars
PSA 230, 2:40-3:30 p.m.
Organizational Meetings
September 6 - Class Field Theory Seminar
PSA 230, 2:40-3:30 p.m.
Ahmed Matar, Central Simple Algebras and Galois Cohomology, Part I
Abstract: Let k be a field. A central simple k-algebra is a simple finite dimensional k-algebra with center k. One of the main reasons these objects are of interest to number theorists is because of their relation to the Brauer group which plays a central role in both local and global class field theory. Classically, this group was defined in terms of central simple algebras. The talk will be over 3 weeks. The first 2 weeks will be devoted to proving some key theorems about central simple algebras and giving both the classical and cohomological description of the Brauer group of a field. In the third week, I will give a mix of results in class field theory using the 2 descriptions of the Brauer group.
September 13 - Number Theory Seminar
ED 212, 12:40-1:30 p.m.
John Jones, An apparent lack of number fields unramified away from 2
Abstract: We will discuss the problem of classifying finite extensions of the rationals which are unramified except at possibly 2 and infinity. Most of the results are negative, ruling out certain types of extensions. For example, new results rule out infinite families of non-abelian simple groups as being the Galois group of such an extension.
September 13 - Class Field Theory Seminar
PSA 230, 2:40-3:30 p.m.
Ahmed Matar, Central Simple Algebras and Galois Cohomology, Part II
September 20 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
John Jones, An apparent lack of number fields unramified away from 2, Part II
September 20 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Ahmed Matar, Central Simple Algebras and Galois Cohomology, Part III
September 27 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Helene Nehrebecki, Quadratic Reciprocity, Part I
October 4 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Helene Nehrebecki, Quadratic Reciprocity, Part II
October 11 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Nancy Childress, The formal groups of Lubin and Tate
Abstract: We introduce the notion of a formal group and discuss several examples, especially those constructed by Lubin and Tate as an approach to local class field theory.
October 18 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Shawn Elledge, Dirichlet Characters, Part I
Abstract: We will discuss Dirichlet Characters. For week one we will cover definitions, basic properties, and examples. For week two we will prove various results, e.g., prime ramification over the associated field of said Dirichlet Character. (Dirichlet Characters are used to define Dirichlet L-functions, an analytic tool used in number theory.)
October 18 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Rachel Wallington, Finding Number Fields with "Small" Galois Root Discriminants, Part I
Abstract: One application of class field theory is to find number fields with solvable Galois groups and Galois root discriminants less than a certain bound. One interesting bound to look at is the Serre-Odlyzko bound which is approximately 44.7. In these talks, I will review some basic ideas about ray class fields, apply these ideas to finding number fields with "small" Galois root discriminant, and show some of my results.
October 25 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Shawn Elledge, Dirichlet Characters, Part II
October 25 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Rachel Wallington, Finding Number Fields with "Small" Galois Root Discriminants, Part II
November 1 - Number Theory Seminar
MOVED TO FRIDAY (THIS WEEK ONLY)
November 1 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Rachel Wallington, Finding Number Fields with "Small" Galois Root Discriminants, Part III
November 2 - Colloquium
PSA 309, 4:00-5:00 p.m.
David Grant, University of Colorado Boulder, Can you hear me now?: the mathematics of space-time codes
Abstract: In recent years, engineers have been concerned with coding for wireless systems with multiple transmit antennas (like cell phone towers) and multiple receive antennas (cell phones, someday). We'll outline some of the ways a surprising amount of combinatorics, algebra, and number theory have been employed in the design of such codes.
November 3 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
David Grant, University of Colorado Boulder, Geometric proofs of reciprocity laws
Abstract: The relationship between power reciprocity laws and geometry has a long history: Eisenstein used the arithmetic of torsion points on the elliptic curves $y^2-y=x^3$ and $y^2=x^3-x$ to give proofs of cubic and biquadratic reciprocity.
Kubota used power reciprocity laws to prove a geometric statement. Namely, let $F$ be an abelian extension of the rationals containing a primitive $\ell^{th}$-root of unity $\zeta$, and let $A$ be an absolutely simple abelian variety over $F$ with complex multiplication $i:{\Cal O_F}\rightarrow\text{End}(A)$ defined over $F$ by the ring of integers ${\Cal O}_F$ of $F$. Then if $t$ is a function in $F(A)$ such that $i(\zeta)^*t=\zeta t$, $\beta\equiv 1\mod{\ell^2}$ is in ${\Cal O}_F$, and $t$ is regular on the non-trivial ${\beta}$-torsion points $A[\beta]'$ of $A$, then
$$\prod_{u\in A[{\beta}]'}t(u)={\frak \beta}^{\sum_{\sigma\in\Phi c_\sigma\sigma}\rho^\ell,\tag 1$$
where $\Phi$ is the CM-type of $(A,i)$, $c_\sigma\in {\Bbb Z}$ is such that $\zeta^{c_\sigma\sigma}=\zeta$, and $\rho\in F$.
We will sketch a proof of the converse: i.e., use formal groups to prove (1) directly for some choices of $A$, $\ell$, $\beta$, and $t$, and then {\it derive} Kummer reciprocity from it. The same approach using a torus with complex multiplication gives Eisenstein reciprocity.
November 8 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Mike McCamy, L-functions, part I
Abstract: L-functions are analytic objects which carry a surprising amount of number theoretic information. I will define them and prove they have an Euler product. I will then discuss the Dedekind zeta function, Dirichlets theorem on primes in arithmetic progression, quadratic residues and class number formulas.
November 8 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Nancy Childress, Formal groups in local class field theory, part I
Abstract: We show how the formal groups constructed by Lubin and Tate can be used to obtain the maximal abelian extension of a local field and the local Artin reciprocity map.
November 15 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Mike McCamy, L-functions, part II
November 15 - Class Field Theory Seminar
LS A 109, 2:40-3:30 p.m.
Nancy Childress, Formal groups in local class field theory, part II
November 22 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Chase Franks, Resolutions of Polynomial Rings and the Hilbert Syzygy, Part I
Abstract: A projective variety V has two invariants, the dimension and degree, which can be computed via Hilbert polynomials. We will see how to compute the Hilbert polynomial of a graded module using a free resolution. The Hilbert Syzygy Theorem tells us that a finitely generated module over k[X_0,...,X_n] has a finite resolution using finitely generated free modules, so that Hilbert polynomials can always be computed by free resolutions. As an application of the functors Tor_n, we will see a succinct proof of the Hilbert Syzygy Theorem. If time permits, some calculations with the program Macaulay 2 will be shown.
November 29 - Number Theory Seminar
PSY B141, 12:40-1:30 p.m.
Chase Franks, Resolutions of Polynomial Rings and the Hilbert Syzygy, Part II
Number Theory Seminar (2005-2006)
