Difference between revisions of "NTS Spring 2013/Abstracts"

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<br>
 
<br>
  
== March 7 ==
+
== February 28 ==
  
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Kai-Wen Lan''' (Minnesota)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Perry''' (NSA)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Galois representations for regular algebraic cuspidal automorphic representations over CM fields
+
| bgcolor="#BCD2EE"  align="center" | Title: The Cracking of Enigma
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: After reviewing what the title means (!) and providing some preliminary explanations, I will report on my joint work with Michael Harris, Richard Taylor, and Jack Thorne on the construction of ''p''-adic Galois representations for regular algebraic cuspidal automorphic representations over CM (or totally real) fields, without hypothesis on self-duality or ramification.  The main novelty of this work is the removal of the self-duality hypothesis; without this hypothesis, we cannot realize the desired Galois representation in the ''p''-adic étale cohomology of any of the varieties we know.  I will try to explain our main new idea without digressing into details in the various blackboxes we need.  I will supply conceptual (rather than technical) motivations for everything we introduce.
+
Abstract: Having learned in the previous talk (Wed., Feb. 27, 5pm–6pm, Van Vleck B239) how the Enigma cryptodevice worked and
|}                                                                       
+
was used by the Germans at the beginning of World War II, we will now learn
</center>
+
precisely how the Polish mathematicians were able to crack the Enigma,
 
+
setting off a series of events that changed the course of world history.
<br>
+
The history of cryptology was also irrevocably changed, with a growing
<!--
+
realization that the future of secrecy would rely on mathematicians and the
== October 11 ==
+
brand new discipline of computer science. This talk is geared towards those
 
+
with some undergraduate mathematics experience, but less is required than
<center>
+
you might suspect.  
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang''' (Madison)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Quaternions and Kudla's matching principle
 
|-
 
| bgcolor="#BCD2EE"
 
Abstract: In this talk, I will explain some interesting identities among average representation numbers by definite quaternions and degree of Hecke operators on Shimura curves (thus indefinite quaternions).  
 
  
 
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<br>
 
<br>
== October 18 ==
 
  
<center>
+
== March 7 ==
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rachel Davis''' (Madison)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: On the images of metabelian Galois representations associated to elliptic curves
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: For ℓ-adic Galois representations associated to elliptic curves, there are theorems
 
concerning when the images are surjective. For example, Serre proved that for a fixed non-CM elliptic curve ''E''/'''Q''', for all but finitely many primes ℓ, the ℓ-adic Galois representation is surjective. Grothendieck and others have developed a theory of Galois representations to an automorphism group of a free pro-ℓ group. In this case, there is less known about the size of the images.
 
The goal of this research is to understand Galois representations to automorphism groups of non-abelian groups more tangibly. Let ''E'' be a semistable elliptic curve over '''Q''' of negative discriminant with good supersingular reduction at 2. Associated to ''E'', there is a Galois representation to a subgroup of the automorphism group of a metabelian group. I conjecture this representation is surjective and give evidence for this conjecture. Then, I compute some conjugacy invariants for the images of the Frobenius elements. This will give rise to new arithmetic information analogous to the traces of Frobenius for the ℓ-adic representation.
 
 
 
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</center>
 
 
 
<br>
 
 
 
== October 25 ==
 
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich''' (Madison)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Kai-Wen Lan''' (Minnesota)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Bounds on characters of SL(2,&nbsp;''q'') via Theta correspondence
+
| bgcolor="#BCD2EE"  align="center" | Title: Galois representations for regular algebraic cuspidal automorphic representations over CM fields
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: I will report on part from a joint project with Roger Howe (Yale).  We develop a method to obtain effective bounds on the irreducible characters of SL(2,&nbsp;''q''). Our method uses explicit realization of all the irreducible representations via the Theta correspondence applied to the dual pair (SL(2,&nbsp;''q''),&nbsp;O), where O is an orthogonal group.
+
Abstract: After reviewing what the title means (!) and providing some preliminary explanations, I will report on my joint work with Michael Harris, Richard Taylor, and Jack Thorne on the construction of ''p''-adic Galois representations for regular algebraic cuspidal automorphic representations over CM (or totally real) fields, without hypothesis on self-duality or ramification. The main novelty of this work is the removal of the self-duality hypothesis; without this hypothesis, we cannot realize the desired Galois representation in the ''p''-adic étale cohomology of any of the varieties we know. I will try to explain our main new idea without digressing into details in the various blackboxes we need. I will supply conceptual (rather than technical) motivations for everything we introduce.
 
 
If you want to learn what are all the notions in my abstract you are welcome to attend the talk. I will not assume any familiarity with the subject.
 
 
|}                                                                         
 
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</center>
 
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<br>
 
<br>
  
== November 1 ==
+
== March 14 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Lei Zhang''' (Boston College)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hang Xue''' (Columbia)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Tensor product L-functions of classical groups of Hermitian type: quasi-split case
+
| bgcolor="#BCD2EE"  align="center" | Title: On the Gan–Gross–Prasad conjecture for U(n)&nbsp;&times;&nbsp;U(n)
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: We study the tensor product L-functions for quasi-split classical groups of Hermitian type times general linear group. When the irreducible automorphic representation of the general linear group is cuspidal and the classical group is an orthogonal group, the tensor product L-functions have been studied by Ginzburg, Piatetski-Shapiro and Rallis in 1997. In this talk, we will show that the global integrals are eulerian and finish the explicit calculation of unramified local L-factors in general.
+
Abstract: In this talk, we shall introduce the Gan–Gross–Prasad conjecture for ''U''(''n'')&nbsp;&times;&nbsp;''U''(''n'') and sketch a proof under certain local conditions using a relative trace formula. We shall also talk about its refinement and applications to the Gan–Gross–Prasad conjecture for ''U''(''n''&thinsp;+1)&nbsp;&times;&nbsp;''U''(''n'').  
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</center>
 
 
 
<br>
 
 
 
== November 8 ==
 
  
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Harron''' (UW–Madison)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Computing Hida families
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: I will report on a joint project with Rob Pollack and four people you know well: Evan Dummit,
 
Marton Hablicsek, Lalit Jain, and Daniel Ross. Our goal is to explicitly compute
 
Hida families using overconvergent modular symbols. This grew out of a
 
project at the Arizona Winter School and the basic idea is to study
 
''p''-adic families of overconvergent modular symbols. I will go over the
 
basic definitions and results starting from classical modular symbols and explain
 
how one goes about encoding these objects on a computer. Aside from
 
being able to compute formal ''q''-expansions of Hida families, we can
 
also compute the structure of the ordinary ''p''-adic Hecke algebra,
 
''L''-invariants, two-variable ''p''-adic ''L''-functions, etc. Several examples
 
will be provided. The code is implemented in Sage.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
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<br>
 
<br>
  
== November 15 ==
+
== April 4 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Lemke Oliver''' (Emory)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''John Jones''' (Arizona State)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Multiplicative functions with small sums
+
| bgcolor="#BCD2EE"  align="center" | Title: The tame-wild principle
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: Analytic number theory is in need of new ideas: for the very
+
Abstract: We consider discriminant relations for number fields, i.e., when the
problem which motivated its existence – the distribution of primes – we
+
discriminant of one field must divide the discriminant of another.  If we
have been unable to make progress in more than fifty years.  Granville and
+
embed the fields in a Galois extension ''L''/''F'' with Galois group ''G'', this
Soundararajan have recently put forward a possible substitute for the
+
can be phrased in terms of subgroups ''H'' and ''K'' of the Galois group: does
seemingly intractable, though admittedly rich, theory of zeros of
+
D_{L^H} | D_{L^K}.  It is easy to prove results of this type under the
''L''-functions.  They dub this new framework the pretentious view of analytic
+
assumption that all ramification is tameWe investigate whether
number theory, where the main objects of consideration are generic
+
consideration of tame ramification is sufficient whether relations which
multiplicative functions, and the goal is to obtain deep theorems about the
+
would always hold for tamely ramified extensions must also hold for wildly
structure of the partial sums of such functionsIn this talk, we consider
+
ramified extensionsWe present successes, failures, and applications (of
multiplicative functions whose partial sums exhibit extreme cancellation.
+
the successes) to computational questions.
We will present two different lines of work about this problem.  First, we
 
develop what might be considered the pretentious framework to answer this
 
question notions of pretentious which permit the detection of power
 
cancellation – which is joint work with Junehyuk Jung of Princeton
 
UniversitySecond, we consider a natural class of functions defined via
 
the arithmetic of number fields, and we classify the members of this class
 
which exhibit extreme cancellation; the proof of this is not at all
 
pretentious.  
 
 
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</center>
 
</center>
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<br>
 
<br>
  
== November 29 ==
+
== April 11 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Xin Shen''' (Minnesota)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Andrew Snowden''' (MIT)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Unramified computation for automorphic tensor ''L''-function
+
| bgcolor="#BCD2EE"  align="center" | Title: Arithmetic families of torsors
|-
 
| bgcolor="#BCD2EE"  |
 
Abstract: In 1967 Langlands introduced the automorphic ''L''-functions and conjectured their analytic properties, including the meromorphic continuation to
 
'''C''' with finitely many poles and a standard functional equation. One
 
of the important cases is the tensor ''L''-functions for ''G''&nbsp;&times;&nbsp;GL<sub>''k''</sub> where
 
''G'' is a classical group. In this seminar I will survey some approaches to this case via integral representations. I will also talk about my recent
 
work on the unramified computation for ''L''-functions of Sp<sub>2''n''</sub>&nbsp;&times;&nbsp;GL<sub>''k''</sub>
 
for the non-generic case.
 
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</center>
 
 
 
<br>
 
 
 
== December 6 ==
 
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Sheng-Chi Liu''' (Texas A&M)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: Subconvexity and equidistribution of Heegner points in the level aspect
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: We will discuss the equidistribution property of Heegner points of level ''q'' and discriminant ''D'', as ''q'' and ''D'' go to infinity. We will establish a hybrid subconvexity bound for certain Rankin–Selberg ''L''-functions which are related to the equdistribution of Heegner points. This is joint work with Riad Masri and Matt Young.
 
 
 
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</center>
 
 
 
<br>
 
 
 
 
 
==<span id="April 16"></span> April 16 (special day: '''Monday''', special time: '''3:30pm–4:30pm''', special place: '''VV B139''') ==
 
 
 
<center>
 
 
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hourong Qin''' (Nanjing U., China)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: CM elliptic curves and quadratic polynomials representing primes
 
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: No polynomial of degree two or higher has been proved to represent infinitely many primes.  Let ''E'' be an elliptic curve defined over '''Q''' with complex multiplication. Fix an integer ''r''.  We  give  sufficient and necessary conditions for ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some prime ''p''. We show that there are infinitely many primes ''p'' such that ''a<sub>p''</sub>&nbsp;=&nbsp;''r'' for some fixed integer ''r'' if
+
Abstract: Let ''G'' be a group scheme over the rational projective line (with some
and only if a quadratic polynomial represents infinitely many primes ''p''.
+
points discarded). Suppose ''X'' is a ''G''-torsor such that ''X<sub>t''</sub> is trivial
 
+
for almost all rational numbers ''t''.  Can we conclude that ''X'' itself is
 +
trivial?  I will discuss several results, some positive and some
 +
negative.  This is joint work with Jacob Tsimerman.
 
|}                                                                         
 
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</center>
 
</center>
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<br>
 
<br>
  
== April 19 ==
+
== April 18 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Robert Guralnick''' (U. Southern California)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jerry Wang''' (Harvard)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: A variant of Burnside and Galois representations which are automorphic
+
| bgcolor="#BCD2EE"  align="center" | Title: Pencils of quadrics and 2-Selmer groups of Jacobians of hyperelliptic curves
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract: Wiles, Taylor, Harris and others used the notion of a big
+
Abstract: Since Bhargava and Shankar's new method of counting orbits, average orders of the 2,3,4,5-Selmer groups of elliptic curves over '''Q''' have been obtained. In this talk we will look at a construction of torsors of Jacobians of hyperelliptic curves using pencils of quadrics and see how they are used to compute the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves over '''Q''' with a rational Weierstrass point.
representation of a finite
 
group to show that certain representations are automorphic.  Jack Thorne
 
recently observed
 
that one could weaken this notion of bigness to get the same conclusions.  He
 
called this property adequate.  An absolutely irreducible  representation ''V''
 
of a finite group ''G'' in characteristic ''p''  is called  adequate  if ''G'' has
 
no ''p''-quotients, the dimension
 
of ''V'' is prime to ''p'', ''V'' has non-trivial self extensions and End(''V'') is
 
generated by the linear
 
span of the elements of order prime to ''p'' in ''G''.     If ''G'' has order
 
prime to ''p'', all of these conditions
 
hold&mdash;the last condition is sometimes called Burnside's Lemma.  We
 
will discuss a recent
 
result of Guralnick, Herzig, Taylor and Thorne showing that if  ''p'' >
 
2 dim ''V'' + 2, then
 
any absolutely irreducible representation is adequate.  We will also
 
discuss some examples
 
showing that the span of the ''p'''-elements in End(''V'') need not be all of End(''V'').
 
 
|}                                                                         
 
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</center>
 
</center>
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<br>
 
<br>
  
== April 26 ==
+
== May 2 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Thorne''' (U. South Carolina)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Wei Ho''' (Columbia/Princeton)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Secondary terms in counting functions for cubic fields
+
| bgcolor="#BCD2EE"  align="center" | Title: Families of lattice-polarized K3 surfaces
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
 +
Abstract: There are well-known explicit families of K3 surfaces equipped with a
 +
low degree polarization, e.g., quartic surfaces in '''P'''<sup>3</sup>.  What if one
 +
specifies multiple line bundles instead of a single one?  We will
 +
discuss representation-theoretic constructions of such families, i.e.,
 +
moduli spaces for K3 surfaces whose Neron–Severi groups contain
 +
specified lattices.  These constructions, inspired by arithmetic
 +
considerations, also involve some fun geometry and combinatorics.
  
Abstract: We will discuss our proof of secondary terms of order ''X''<sup>5/6</sup> in the Davenport–Heilbronn
+
This is joint work with Manjul Bhargava and Abhinav Kumar.  
theorems on cubic fields and 3-torsion in class groups of quadratic fields. For cubic
 
fields this confirms a conjecture of Datskovsky–Wright and Roberts. We also will describe
 
some generalizations, in particular to arithmetic progressions, where we discover a
 
curious bias in the secondary term.
 
 
 
Roberts’ conjecture has also been proved independently by Bhargava, Shankar, and
 
Tsimerman. Their proof uses the geometry of numbers, while our proof uses the analytic
 
theory of Shintani zeta functions.
 
 
 
We will also discuss a combined approach which yields further improved error terms. If
 
there is time (or after the talk), I will also discuss a couple of side projects and my
 
plans for further related work.
 
 
 
This is joint work with Takashi Taniguchi.
 
 
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
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<br>
 
<br>
  
== May 3 ==
+
== May 9 ==
  
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Alina Cojocaru''' (U. Illinois at Chicago)
+
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Stephen Gelbart''' (Weizmann Institute)
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Frobenius fields for elliptic curves
+
| bgcolor="#BCD2EE"  align="center" | Title: A ''p''-adic integral for the reciprocal of the ''p''-adic ''L''-function ''L''(''s'',&nbsp;χ)
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| bgcolor="#BCD2EE"  |
Abstract: Let ''E'' be an elliptic curve defined over '''Q'''. For a prime p of good reduction for ''E'', let &pi;<sub>p</sub> be the p-Weil root of E and '''Q'''(&pi;<sub>p</sub>) the associated imaginary quadratic field generated by &pi;<sub>p</sub>. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes ''p < x'' for which '''Q'''(&pi;<sub>p</sub>) is isomorphic to a fixed imaginary quadratic field. I will discuss progress on this conjecture, in particular an average result confirming the predicted asymptotic formula. This is joint work with Henryk Iwaniec and Nathan Jones. 
+
Abstract: We introduce an analog of part of the Langlands–Shahidi method to the ''p''-adic setting, constructing reciprocals of certain ''p''-adic ''L''-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group ''SL''(2), and give explicit ''p''-adic measures whose Mellin transforms are reciprocals of Dirichlet ''L''-functions.
 
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
 
<br>
 
<br>
 
== May 10 ==
 
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Samit Dasgupta''' (UC Santa Cruz)
 
|-
 
| bgcolor="#BCD2EE"  align="center" | Title: The ''p''-adic ''L''-functions of evil Eisenstein series
 
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Abstract: Let ''f'' be a newform of weight ''k''+2 on &Gamma;<sub>1</sub>(''N''), and let ''p''&nbsp;∤&nbsp;''N'' be a prime. For each root &alpha; of the Hecke polynomial of ''f'' at ''p'', there is a corresponding ''p''-stabilization ''f''<sub>&alpha;</sub> on &Gamma;<sub>1</sub>(''N'')&nbsp;∩&nbsp;&Gamma;<sub>0</sub>(''p'') with ''U<sub>p''</sub>-eigenvalue equal to &alpha;. The construction of ''p''-adic ''L''-functions associated to such forms ''f''<sub>&alpha;</sub> has been much studied. The non-critical case (when ord<sub>''p''</sub>(&alpha;)&nbsp;<&nbsp;''k''+1) was handled in the 1970s via interpolation of the classical ''L''-function in work of Mazur, Swinnerton-Dyer, Manin, Visik, and Amice–V&eacute;lu.  Recently, certain critical cases were handled by Pollack and Stevens, and the remaining cases were finished off by Bellaïche. Many years prior to Bellaïche's proof of their existence, Stevens had conjectured a factorization formula for the ''p''-adic ''L''-functions of evil (i.e. critical) Eisenstein series based on computational evidence. In this talk we describe a proof of Stevens's factorization formula. A key element of the proof is the theory of distribution-valued partial modular symbols.  This is joint work with Joël Bellaïche. 
 
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Latest revision as of 14:09, 25 April 2013

January 24

Tamar Ziegler (Technion)
Title: An inverse theorem for the Gowers norms

Abstract: Gowers norms play an important role in solving linear equations in subsets of integers - they capture random behavior with respect to the number of solutions. It was conjectured that the obstruction to this type of random behavior is associated in a natural way to flows on nilmanifolds. In recent work with Green and Tao we settle this conjecture. This was the last piece missing in the Green-Tao program for counting the asymptotic number of solutions to rather general systems of linear equations in primes.


January 31

William Stein (U. of Washington)
Title: How explicit is the explicit formula?

Abstract: Consider an elliptic curve E. The explicit formula for E relates a sum involving the numbers ap(E) to a sum of three quantities, one involving the analytic rank of the curve, another involving the zeros of the L-series of the curve, and the third, a bounded error term. Barry Mazur and I are attempting to see how numerically explicit – for particular examples – we can make each term in this formula. I'll explain this adventure in a bit more detail, show some plots, and explain what they represent.

(This is joint work with Barry Mazur).


February 7

Nigel Boston (Madison)
Title: A refined conjecture on factoring iterates of polynomials over finite fields

Abstract: In previous work Rafe Jones and I studied the factorization of iterates of a quadratic polynomial over a finite field. Their shape has consequences for the images of Frobenius elements in the corresponding Galois groups (which act on binary rooted trees). We found experimentally that the shape of the factorizations can be described by an associated Markov process, we explored the consequences to arboreal Galois representations, and conjectured that this would be the case for every quadratic polynomial. Last year I gave an undergraduate, Shixiang Xia, the task of accumulating more evidence for this conjecture and was shocked since one of his examples behaved very differently. We have now understood this example and come up with a modified model to explain it.


February 14

Tonghai Yang (Madison)
Title: A high-dimensional analogue of the Gross–Zagier formula

Abstract: In this talk, I will explain roughly how to extend the well-known Gross–Zagier formula to unitary Shimura varieties of type (n − 1, 1). This is a joint work with J. Bruinier and B. Howard.


February 28

David Perry (NSA)
Title: The Cracking of Enigma

Abstract: Having learned in the previous talk (Wed., Feb. 27, 5pm–6pm, Van Vleck B239) how the Enigma cryptodevice worked and was used by the Germans at the beginning of World War II, we will now learn precisely how the Polish mathematicians were able to crack the Enigma, setting off a series of events that changed the course of world history. The history of cryptology was also irrevocably changed, with a growing realization that the future of secrecy would rely on mathematicians and the brand new discipline of computer science. This talk is geared towards those with some undergraduate mathematics experience, but less is required than you might suspect.


March 7

Kai-Wen Lan (Minnesota)
Title: Galois representations for regular algebraic cuspidal automorphic representations over CM fields

Abstract: After reviewing what the title means (!) and providing some preliminary explanations, I will report on my joint work with Michael Harris, Richard Taylor, and Jack Thorne on the construction of p-adic Galois representations for regular algebraic cuspidal automorphic representations over CM (or totally real) fields, without hypothesis on self-duality or ramification. The main novelty of this work is the removal of the self-duality hypothesis; without this hypothesis, we cannot realize the desired Galois representation in the p-adic étale cohomology of any of the varieties we know. I will try to explain our main new idea without digressing into details in the various blackboxes we need. I will supply conceptual (rather than technical) motivations for everything we introduce.


March 14

Hang Xue (Columbia)
Title: On the Gan–Gross–Prasad conjecture for U(n) × U(n)

Abstract: In this talk, we shall introduce the Gan–Gross–Prasad conjecture for U(n) × U(n) and sketch a proof under certain local conditions using a relative trace formula. We shall also talk about its refinement and applications to the Gan–Gross–Prasad conjecture for U(n +1) × U(n).


April 4

John Jones (Arizona State)
Title: The tame-wild principle

Abstract: We consider discriminant relations for number fields, i.e., when the discriminant of one field must divide the discriminant of another. If we embed the fields in a Galois extension L/F with Galois group G, this can be phrased in terms of subgroups H and K of the Galois group: does D_{L^H} | D_{L^K}. It is easy to prove results of this type under the assumption that all ramification is tame. We investigate whether consideration of tame ramification is sufficient – whether relations which would always hold for tamely ramified extensions must also hold for wildly ramified extensions. We present successes, failures, and applications (of the successes) to computational questions.


April 11

Andrew Snowden (MIT)
Title: Arithmetic families of torsors

Abstract: Let G be a group scheme over the rational projective line (with some points discarded). Suppose X is a G-torsor such that Xt is trivial for almost all rational numbers t. Can we conclude that X itself is trivial? I will discuss several results, some positive and some negative. This is joint work with Jacob Tsimerman.


April 18

Jerry Wang (Harvard)
Title: Pencils of quadrics and 2-Selmer groups of Jacobians of hyperelliptic curves

Abstract: Since Bhargava and Shankar's new method of counting orbits, average orders of the 2,3,4,5-Selmer groups of elliptic curves over Q have been obtained. In this talk we will look at a construction of torsors of Jacobians of hyperelliptic curves using pencils of quadrics and see how they are used to compute the average order of the 2-Selmer groups of Jacobians of hyperelliptic curves over Q with a rational Weierstrass point.


May 2

Wei Ho (Columbia/Princeton)
Title: Families of lattice-polarized K3 surfaces

Abstract: There are well-known explicit families of K3 surfaces equipped with a low degree polarization, e.g., quartic surfaces in P3. What if one specifies multiple line bundles instead of a single one? We will discuss representation-theoretic constructions of such families, i.e., moduli spaces for K3 surfaces whose Neron–Severi groups contain specified lattices. These constructions, inspired by arithmetic considerations, also involve some fun geometry and combinatorics.

This is joint work with Manjul Bhargava and Abhinav Kumar.


May 9

Stephen Gelbart (Weizmann Institute)
Title: A p-adic integral for the reciprocal of the p-adic L-function L(s, χ)

Abstract: We introduce an analog of part of the Langlands–Shahidi method to the p-adic setting, constructing reciprocals of certain p-adic L-functions using the nonconstant terms of the Fourier expansions of Eisenstein series. We carry out the method for the group SL(2), and give explicit p-adic measures whose Mellin transforms are reciprocals of Dirichlet L-functions.



Organizer contact information

Robert Harron

Zev Klagsbrun

Sean Rostami


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