Schedule of Talks
The talks will take place in Lecture Center D-2 on the UIC East Campus. To get to campus, take the Blue Line train to UIC-Halsted, or walk to the intersection of Harrison and Halsted.
Here are more detailed directions.
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| Friday, March 7, 2008
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March 7
2pm - 3pm |
REFRESHMENTS
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March 7
3pm - 3:50pm |
Winnie Li
(Penn State)
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Noncongruence modular forms and modularity
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March 7
3:50pm - 4:10pm |
BREAK
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March 7
4:10pm - 4:30pm |
Steven Spallone
(Purdue)
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P-adic local dynamics
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March 7
4:30pm - 4:40pm |
BREAK
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March 7
4:40pm - 5pm |
Mehmet Haluk Sengun
(Wisconsin)
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Galois representations of small quadratic fields
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March 7
5pm - 7pm |
RECEPTION AT UIC
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March 7
7:30pm |
DINNER in GREEK TOWN
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| Saturday, March 8, 2008
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March 8
9:30am - 10am |
REFRESHMENTS
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March 8
10am - 10:50am |
Wee Teck Gan
(UC San Diego)
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Formal degree and local theta correspondence
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March 8
10:50am - 11:10am |
BREAK
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March 8
11:10am - 11:30am |
Ghaith Hiary
(Minnesota)
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Fast methods to compute the Riemann zeta function
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March 8
11:10am - 11:40am |
BREAK
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March 8
11:40am- 12pm |
Ling Long
(Iowa State)
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On Atkin and Swinnerton-Dyer congruences of some noncongruence
cuspforms
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March 8
12pm - 2pm |
LUNCH
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March 8
2pm - 2:50pm |
Noam Elkies
(Harvard)
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How many points can a genus-2 curve have?
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March 8
2:50pm - 3:10pm |
BREAK
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March 8
3:10pm - 3:30pm |
Nick Ramsey
(Michigan)
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The overconvergent Shimura lifting
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March 8
3:30pm - 3:40pm |
BREAK
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March 8
3:40pm - 4pm |
Maosheng Xiong
(Penn State)
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The Erdos-Kac theorem for polynomials of several variables
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March 8
4pm - 4:10pm |
BREAK
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March 8
4:10pm - 5pm |
Dinesh Thakur
(Arizona)
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Nature of Gamma, Zeta and Multizeta values in function field arithmetic
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March 8
5pm - 6pm |
CLOSING RECEPTION AT UIC
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March 8
6pm - 8pm |
DINNER
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| End
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ABSTRACTS
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Winnie Li,
Noncongruence modular forms and modularity
Unlike congruence modular forms, the arithmetic of
noncongruence modular forms is not much understood, due to
the lack of Hecke operators. For noncongruence forms, Atkin and
Swinnerton-Dyer proposed a substitute of the familiar degree two recursive
relation satisfied by congruence Hecke eigenforms by three-term
congruence relations. In a certain situation, this yields very interesting
congruence relations between the Fourier coefficients of congruence and
noncongruence forms.
In this survey talk, we shall review the development of noncongruence
forms, discuss the progress on congruence relations, as well as the
unbounded denominator conjecture, which asserts that the algebraic
noncongruence forms are distinguished by its Fourier coefficients having
unbounded denominators. We shall see that the modularity of certain Galois
representations plays an essential role.
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Steven Spallone, P-adic local dynamics
We determine conjugacy classes of p-adic analytic germs of power
series tangent to the identity. This is joint work with Adrian Jenkins.
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Mehmet Haluk Sengun, Galois representations of small quadratic fields
I will present the result that for small quadratic fields K and
primes p, there is no irreducible continuous mod p representation of the
absolute Galois group of K that is unramified away from p and infinity. I
will discuss Serre's conjecture and its generalizations as a motivation to
this result.
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Wee Teck Gan,
Formal degree and local theta correspondence
The formal degree of a discrete series representation of a
reductive group over a local field is the analog of the dimension of a
finite-dimensional representation. A recent conjecture of
Hiraga-Ichino-Ikeda expresses the formal degree of \pi in terms of the
adjoint gamma factor of the Langlands parameter of \pi. I will explore the
connection of this conjecture with Langlands functoriality and theta
correspondence, and if time permits, discuss the case of GSp(4).
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Ghaith Hiary, Fast methods to compute the Riemann zeta function
The Riemann zeta function on the critical line can be computed using
a straightforward application of the Riemann-Siegel formula, Sch\"onhage's
method, or Heath-Brown's method. The complexities of these methods have
exponents 1/2, 3/8 (=0.375), and 1/3 respectively. In this talk, two new fast
and potentially practical methods to compute zeta are presented. One method
relies on an algorithm to compute quadratic exponential (theta) sums. Its
complexity has exponent 1/3. The second method employs an algorithm to compute
cubic exponential sums. Its complexity has exponent 4/13 (approximately,
0.307). If time permits, I will also present the results of recent computations
(with Andrew Odlyzko) of moments and other statistics of zeta. The computations
were done for a set of 20*10^9 zeros near the zero 10^23, as well as at lower
heights.
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Ling Long, On Atkin and Swinnerton-Dyer congruences of some noncongruence
cuspforms
Serious investigations on modular forms for noncongruence
subgroups of the modular group were pioneered by Atkin and Swinnerton. They
discovered numerically the so-called three-term Atkin and Swinnerton-Dyer
congruences satisfied by the Fourier coefficients of noncongruence modular
forms. There are $p$-adic analogues of the three-term Hecke recursions
satisfied by the classical congruence Hecke eigenforms.
In this talk we will show that the Atkin and Swinnerton-Dyer congruences are
satisfied by some special weight 4 noncongruence cuspforms. This is a joint
work with A.O.L. Atkin.
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Noam Elkies, How many points can a genus-2 curve have?
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Nick Ramsey, The overconvergent Shimura lifting
I'll discuss a rigid-analytic interpolation of the Shimura
lifting across the eigencurve and consequences for interpolation of L
values.
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Maosheng Xiong, The Erdos-Kac theorem for polynomials of several variables
We prove two versions of the Erdos-Kac theorem for polynomials of
several variables on some varieties arising from translation and affine linear
transformation.
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Dinesh Thakur, Nature of Gamma, Zeta and Multizeta values in function field arithmetic
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