Thursday, January 17, 2 p.m., Room 241 Altgeld Hall, Prof. Nigel Boston,
Organizational meeting.
Thursday, January 24, 2 p.m., Room 241 Altgeld Hall, Prof. Marcin Mazur,
"Galois module structure of units in real biquadratic number fields".
Thursday, January 31, 2 p.m., Room 241 Altgeld Hall, Prof. Scott Ahlgren,
"Weierstrass points on modular curves and supersingular j-invariants".
Thursday, February 7, 2 p.m., Room 241 Altgeld Hall, Prof. Nigel Boston,
"Selberg's conjectures".
Abstract: Dirichlet series in Selberg's family S admit analytic continuation,
Euler products, functional equations, and bounded growth
of their coefficients. S contains all Dirichlet, Hecke, and
conjecturally Artin L-functions. It is also a monoid under multiplication
and one can ask if it has unique factorization into
"primitive" series. Selberg's conjectures that the
primitive series in S satisfy a sort of orthonormality,
indeed imply unique factorization and much more. The
talk will give an introduction to these topics.
Thursday, February 14, 2 p.m., Room 241 Altgeld Hall, Bogdan Petrenko,
"On the product of two primitive elements of maximal
subfields of a finite field".
Abstract. Let $K$ be a finite field and $K(a), K(b)$
finite dimensional extensions that are maximal subfields of $K(a,b)$.
We investigate when $K(a,b) = K(ab)$ provided
$(K(a,b):K)$ is the product of three distinct primes.
We employ Galois theory, group theory, and
elementary number theory.
Thursday, February 21, 2 p.m., Room 241 Altgeld Hall, Prof. William McGraw (Wisconsin),
"The Rationality of Vector Valued Modular Forms and Implications for a
Theorem of Borcherds of Gross-Kohnen-Zagier Type".
Abstract: In a recent paper,
Borcherds generalizes a theorem of Gross, Kohnen, and Zagier
by showing that the higher
dimensional analogues of Heegner divisors are related to the
coefficients of certain vector valued modular forms associated to the
Weil representation. Borcherds notes that this relationship can be made
more explicit if it can be shown that the spaces of these modular forms
have bases whose Fourier expansions have only integer coefficients. We
show that this is the case, allowing us to state a stronger and simpler
version of Borcherds' main theorem.
Friday, March 1, 3 p.m., Room 140 Henry Building, Prof. Harold Edwards (Courant Institute of Mathematical Sciences),
"Gauss's Second Proof of Quadratic Reciprocity".
Abstract: Gauss's Disquisitiones Arithmeticae contains two proofs of the
law of quadratic reciprocity. The second uses Gauss's forbidding theory
of composition of quadratic forms. The talk will sketch a simplification
of this theory and show how it can be applied to the proof of quadratic
reciprocity.
Thursday, March 7, 2 p.m., Room 241 Altgeld Hall, Prof. Dr. Peter Schneider (University of Muenster, Germany),
"Representation theory and algebras of p-adic distributions".
Abstract: Together with J. Teitelbaum we pursue a long term program to
develop the representation theory of p-adic reductive groups in p-adic
vector spaces and its connection to p-adic Galois representations. As
always in representation theory group representations are studied as
modules over an appropriate group ring. In our context this is the
convolution algebra of locally analytic p-adic distributions on the group.
These algebras are not noetherian and not much is known about their
structure. I will explain some ideas how to construct a well behaved
abelian category of "coherent" modules over these algebras and how it
relates to group representations.
Thursday, March 14, 2 p.m., Room 241 Altgeld Hall, Prof. Joshua Holden (Rose-Hulman Institute of Technology), "Counting Fontaine-Mazur-like function fields".
Abstract: If F is a number field and l is a prime, Fontaine and Mazur
conjectured that there are no infinite unramified extensions of F
with l-adic analytic Galois groups. Does the same hold true if F is
a function field? In general it does not, but we suspect that
violations of this property are rather rare. One can put conditions
on F which force the property to hold; these conditions may be
formulated in terms of the action of Frobenius. Jeff Achter and I
then used equidistribution results for l-adic monodromy to make
certain asymptotic counts of the fields involved. These show that
most function fields obey the property, although the question is
probably still undecided for a positive proportion.
Thursday, March 21. No meeting - spring break.
Thursday, March 28, 2 p.m., Room 241 Altgeld Hall, Prof. Adrian Iovita (Univ. of Washington),
"p-Adic families of exponential maps attached to families of modular forms".
Abstract:The exponential maps are generalizations of the so called
"Coleman power series" maps. They are maps which connect
p-adic L-functions to Euler systems, both attached to global objects like
elliptic curves or modular forms. We show that the exponential maps
attached to a p-adic family of modular forms live in a p-adic analytic
family. We expect that this will have interesting arithmetic consequences.
Thursday, April 11, 2 p.m., Room 241 Altgeld Hall, Michael Bush (UIUC),
"Schur Multipliers and Number Theory".
Abstract:We give an elementary description of the Schur multiplier of a finite group and describe some of its properties. A
simple application in algebraic number theory will be presented.
Tuesday, April 16, 1 p.m., Room 243 Altgeld Hall, James McLaughlin (UIUC),
(joint seminar), "Small Prime Powers in the Fibonacci Sequence".
Abstract:Using elementary means, Cohn and Wylie proved independently in 1964 that the only squares in the Fibonacci
sequence are $F_{0}=0,F_{1}= F_{2} = 1$ and $F_{12} = 144$. London and Finkelstein used previous results on solutions to two
diophantine equations to show in 1970 that the only cubes in the Fibonacci sequence are $F_{0}=0$, $F_{1}= F_{2} = 1$ and $F_{6} =
8$. In 1983, Peth\H{o} used linear forms in logarithms together with a computer search using congruence considerations to give an
alternative proof of London and Finkelstein's result and also states that he used the same methods to show that only fifth powers in the
Fibonacci sequence are $F_{0}=0$ and $F_{1}= F_{2} =1$. Here the method outlined by Peth\H{o} is used as a starting point and then
linear forms in logarithms together with the LLL algorithm are used to reprove the result for fifth powers and to prove that the only
seventh-, eleventh-, thirteenth- or seventeenth powers in the Fibonacci sequence are $F_{0}=0$ and $F_{1}= F_{2} =1$. An
alternative method (to the usual exhaustive check of products of powers of fundamental units) is used to complete the search in the case of
eleventh, thirteenth- and seventeenth powers.
Thursday, April 18, 2 p.m., Room 241 Altgeld Hall, Prof. Leon McCulloh (UIUC),
"Generalizing the Iwasawa-Sinnott class number formula
from cyclotomic fields to integral group rings".
Abstract: Iwasawa reinterpreted the analytic class number
formula for the first ("minus") factor of the class number of
the field of (p^n)th roots of unity as the order of the torsion
subgroup of the quotient of the Galois group ring by a
Stickelberger ideal. Sinnott generalized this result to an
arbitrary cyclotomic field. It can be further generalized
to the class number of some integral group rings ZG, relating
it to the order of the torsion subgroup of the quotient of
ZG by a Stickelberger-like submodule derived from the character
table of the G.
Tuesday, April 23, 1 p.m., Room 243 Altgeld Hall, Prof. David Grant (University of Colorado),
(joint seminar), "Just how exceptional are exceptional primes?".
Abstract: Let $E$ be an elliptic curve defined over $\Bbb Q$, and $p$ be a prime.
We say $p$ is exceptional for $E$ if the Galois representation on $E[p]$
is not a full $Gl_2({\Bbb Z}/p{\Bbb Z})$ extension. Every elliptic curve
over $\Bbb Q$ has a unique model of the form $y^2=x^3+Ax+B,$ with $A,B\in
{\Bbb Z}$, which is minimal in the sense that $gcd(A^3,B^2)$ is
twelfth-power free, and we define the naive height $H(E)$ to be
$max(|A^3|,B^2)$. If ${\Cal C}(X)$ is the set of elliptic curves $E$ with
$H(E)\leq X^6$, then $|{\Cal C}(X)|=O(X^5)$. Let ${\Cal E}(X)$ be the
curves in ${\Cal C}(X)$ with at least one exceptional primes. We will
prove a conjecture of Duke that $|{\Cal E}(X)| \sim C X^3,$ for some
constant $C$.
Thursday, April 25, 1 p.m., Room 243 Altgeld Hall, Prof. Matt Papanikolas (Brown Univ.),
(joint seminar), "Drinfeld modules and extensions".
Abstract: For an abelian variety A, the group of extensions of A by the multiplicative group can be given the structure of an abelian variety. The Weil-Barsotti formula provides a
natural identification of this abelian variety with the dual of A. In a similar direction, we consider extensions of Drinfeld modules by Drinfeld modules. We find an analogue of the Weil-Barsotti
formula (by extending by the Carlitz module) as well as other results. Joint with N. Ramachandran.
Thursday, April 25, 2 p.m., Room 241 Altgeld Hall, Prof. Franz Lemmermeyer (CSUSM),
"Class groups of dihedral extensions."
Abstract: In 1974, Callahan conjectured that in a dihedral
extension $L/Q$ with quadratic subfield $k$ and
cubic subfield $K$ we have the relation
$r(k) = r(K)+1$ between the ranks of the
$p$-class groups of $k$ and $K$ if $L/k$ is
unramified. This was later proved independently by
Gerth and Gras, and extended to complex dihedral
extensions of degree 2p over $Q$ by Boelling. Here
I want to show how to generalize this relation to
arbitrary dihedral extensions of degree $2p$ and
arbitrary base fields of class number not divisible
by $p$.
Volunteers and those with suggestions, please contact n-boston@uiuc.edu.
Note also our Langlands theory seminar.