RAP: Number Theory and Modular Forms
Tuesday, 2-3 pm,
347 Altgeld
Friday, 4-5 pm,
241 Altgeld
University of Illinois -
Spring 2004
Organizers:
Scott Ahlgren (ahlgren@math.uiuc.edu)
Matt Boylan (boylan@math.uiuc.edu).
Description:
We will consider connections between modular forms, Galois representations,
and problems in Number Theory. Participants will give talks on
background material and on current research papers. Pace and precise
material to be covered will by determined by the interests of the
participants.
Schedule of Talks:
Friday, Jan. 30: Ricardo Rojas:
The Galois group of \overbar(Q)/Q.
Tuesday, Feb. 3: Ricardo Rojas:
The Galois group of \overbar(Q)/Q (continued).
Reference:
| [1] |
J. Neukirch,
Algebraic Number Theory,
Springer, Grundlehren der mathematischen Wissenschaften, 322,
(1992), Chapter IV, p.261-274. |
Friday, Feb. 6: Barry Walker:
Introduction to representations of the Galois group of \overbar(Q)/Q, part I.
References:
| [2] |
H. P. F. Swinnerton-Dyer,
On $\ell$-adic representations and congruences for coefficients of
modular forms, in
Modular Functions of One Variable III, W. Kuyk and
J.-P. Serre eds. Springer Lect. Notes in Math. 350, (1973) 1-55
(especially sections 2 and 4). |
| [3] |
S. Lang,
Introduction to Modular Forms,
Springer, Grundlehren der mathematischen Wissenschaften, 222,
(1976), Chapter XI. |
Tuesday, Feb. 10: Barry Walker:
Introduction to representations of the Galois group of \overbar(Q)/Q,
part II.
Friday, Feb. 13: Barry Walker:
Introduction to representations of the Galois group of \overbar(Q)/Q,
part III.
Tuesday, Feb. 17: Scott Ahlgren:
Examples of congruences for the tau function.
Reference:
| [4] |
J.-P. Serre,
Une interpr\'etation des congruences relatives \`a la fonction $\tau$
de Ramanujan,
S\'em. Delange-Pisot-Positou (Th\'eorie des nombres) (1967/68),
no. 14.
|
Friday, Feb. 20: Timothy Kilbourn:
Divisibility of integer weight modular form coefficients.
Reference:
| [5] |
J.-P. Serre,
Divisibilit\'e des coefficients des formes modulaires de poids entier,
C. R. Acad. Paris, 279, s\'erie A (1974), 679-682
=Oeuvres Vol. 3, 100, 189-193. |
Tuesday, Feb. 24: Timothy Kilbourn:
Divisibility of integer weight modular form coefficients, part II.
Friday, Feb. 27: We will be discussing
problems 6.3, 6.5, 6.6, and 6.7 from the following paper of Serre.
Reference:
| [6] |
J.-P. Serre,
Divisiblit\'e des certains fonctions arithm\'etiques,
L'Ens. Math. 22, (1976), 227-260
=Oeuvres Vol. 3, 108, 250-283. |
Tuesday, March 2: Stephanie Treneer:
Distribution of the partition function modulo $m$.
References:
| [7] |
S. Ahlgren and K. Ono,
Congruence properties for the partition function,
Proc. Nat. Acad. Sci. USA, 98, Issue 23, (2001), 12882-12884. |
| [8] |
K. Ono
The web of modularity: Arithmetic of the coefficients of modular forms
and $q$-series,
Amer. Math. Soc., CBMS Regional Conf. in Math., vol 102 (2004),
section 5.2.4, pages 97-101. |
A related paper by K. Ono is
| [9] |
K. Ono,
Distribution of the partition function modulo $m$,
Ann. of Math. 151, (2000), 293-307. |
Friday, March 5: Stephanie Treneer:
Distribution of the partition function modulo $m$, part II.
Tuesday, March 9: Matt Boylan:
Introduction to modular forms with complex multiplication.
Friday, March 12: Matt Boylan:
Complex multiplication and lacunarity of integer weight forms.
References:
| [10] |
K. Ribet,
Galois representations attached to eigenforms with Nebentypus,
Springer Lect. Notes in Math., 601, (1977), 17-52.
|
| [11] |
J.-P. Serre,
Quelques applications du th\'eor\`eme de densit\'e de Chebotarev,
Publ. Math. I.H.E.S., 54, (1981), 123-201
=Oeuvres, Vol. 3, 125, 563-641.
|
| [12] |
J.-P. Serre,
Sur la lacunarit\'e des puissances de $\eta$,
Glasgow Math. J., 27, (1985), 203-221
=Oeuvres, Vol. 4, 139, 66-84.
|
Tuesday, March 16: Ricardo Rojas:
Introduction to Gaussian hypergeometric series.
Friday, March 19: Ricardo Rojas:
Introduction to Gaussian hypergeometric series, II.
Reference:
| [13] |
K. Ono,
Values of Gaussian hypergeometric series,
Trans. Amer. Math. Soc., 350, no. 3, (Mar. 1998), 1205-1223.
|
Tuesday, April 6, 2-3, 347 Altgeld Hall:
Iwan Duursma:
Algebraic definitions for curves, their Jacobian, elliptic curves, and
abelian varieties over arbitrary ground fields.
Thursday, April 8, 2-3, 241 Altgeld Hall:
Timothy Kilbourn:
Analytic definitions for curves as Riemann surfaces, their Jacobian,
module of differentials, Abel-Jacobi theorem. Applications to modular curves
and modular symbols.
Tuesday, April 13, 2-3, 347 Altgeld Hall:
Iwan Duursma: More on modular curves and abelian varieties.
We pick up two objects that were defined analytically
by Tim last Thursday: the modular curve H*/Gamma0(N)
and the abelian variety C^g/Lambda. We aim for a
purely algebraic description of the function field
for the modular curve of level N=2^n, any n.
Thursday, April 15, 2-3,
241 Altgeld Hall: Nadia Masri: Modular curves of level $11$.
We give some explicit examples
of general theory using the curves X_0(11) and X_1(11).
Reference:
Friday, April 16, 4 pm,
241 Altgeld Hall: William Stein (Harvard): Visibility of
Shafarevich-Tate groups of modular abelian varieties at higher level.
(This is in conjunction with the regularly scheduled number theory seminar.)
Abstract:
I will begin by introducing the Birch and Swinnerton-Dyer conjecture in the
context of abelian varieties attached to modular forms, and discuss some of
the main results about it. I will then introduce Mazur's notion of visibility
of Shafarevich-Tate groups and explain some of the basic facts and theorems.
Cremona, Mazur, Agashe, and myself carried out large computations about
visibility for modular abelian varieties of level N in J_0(N). These
computations addressed the following question: If A is a modular abelian
variety of level N, how much of the Shafarevich-Tate group of A is modular
of level N, i.e., visible in J_0(N). The results of these computations
suggest that often much of the Shafarevich-Tate group is not modular of
level N. This suggests asking if every element is modular of level N*m, for
some auxiliary integer m, and if so, what can one say about the set of
such m? I will finish the talk with some new data and thoughts about this
last question, which is still very much open.
Tuesday, April 20, 1 pm,
241 Altgeld Hall: Akshay Venkatesh (MIT): Uniform bounds for the
number of points on curves.
(This is in conjunction with the regularly scheduled number theory seminar.)
Abstract:
A lovely result of Bombieri and Pila states that any irreducible
curve of degree d in the plane passes through
at most N^{1/d+epsilon} lattice points in a box of size N.
Heath-Brown proved a corresponding result for rational points.
In both cases, the remarkable feature is the uniformity in the curve;
unfortunately, they do not detect genus, i.e. they say nothing more
about a curve of genus 1 than a rational curve.
I'll explain how these uniform results are proved.
I'll then discuss an approach that gives slightly better results for genus
> 0 and apply it to bounding 3-torsion in quadratic class groups.
This is a combination of joint work with Helfgott and with Ellenberg.
Thursday, April 22, 2 pm, 241 Altgeld Hall
: Jennifer Paulhus: On the Modularity of Fermat curves.
Abstract:
Abstract: For all positive integers d not equal to 1, 2, 3, 4, 6, 8 or 12 it
is known that the Fermat Curve Fd: x^d+y^d+z^d=0 is not modular. For
d=1, 2, 3, 4 and 8, Fd is known to be modular. In this talk we use Gauss and
Jacobi Sums to determine which elliptic curves occur in the decomposition of
the Jacobian of F6. The answer to this should help us determine whether F6 is
modular.
Tuesday, April 27, 1 pm,
241 Altgeld Hall: Robert Pollack (University of Chicago):
Elliptic curves and Iwasawa Theory.
(This is in conjunction with the regularly scheduled number theory seminar.)
Abstract:
The set of rational solutions on an elliptic curve forms an abelian group
and (by a famous theorem of Mordell) this group is actually finitely
generated. Determining the rank of the these groups remains today as a major
open question in number theory. It is also true that the set of K-valued
solutions of an elliptic curve is finitely generated whenever K is some
finite extension of the rationals numbers. One may ask how these ranks
vary as K varies through certain towers of number fields. Questions relating
to the behavior of such algebraic invariants in towers forms the basis of
Iwasawa theory of elliptic curves. In this talk, we will review the basics
of the arithmetic of elliptic curves and their Iwasawa theory. Moreover,
we will discuss some recent developments of this theory in the
supersingular case.