Difference between revisions of "NTS Fall 2012/Abstracts"

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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yongqiang Zhao''' (Madison)
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| bgcolor="#BCD2EE"  align="center" | Title: On the Roberts conjecture
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Abstract: Let ''N''(''X'')&nbsp;=&nbsp;#{''K''&nbsp;<nowiki>|</nowiki>&nbsp;[''K'':'''Q''']&nbsp;=&nbsp;3,&nbsp;disc(''K'')&nbsp;≤&nbsp;''X''} be the counting function of cubic fields of bounded
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Abstract: tba
discriminant. The Roberts Conjecture is about the second term of this counting function. This conjecture was thought to be hard and dormant for some time. However, recently, four very different
 
approaches to this problem were developed independently by Bhargava, Shankar and Tsimerman,
 
Hough, Taniguchi and Thorne, and myself.
 
In this talk, I will mention the first three approaches very briefly, then focus on my results on the function field case. I will give an outline of the proof. If time permits, I will indicate how the geometry
 
feeds back to the number field case, in particular, how one could possibly define a new invariant
 
for cubic fields.
 
 
 
 
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== March 22 ==
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Paul Terwilliger''' (Madison)
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Lei Zhang''' (Boston College)
 
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| bgcolor="#BCD2EE"  align="center" | Title: Introduction to tridiagonal pairs
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| bgcolor="#BCD2EE"  align="center" | Title: Tensor product L-functions of classical groups of Hermitian type: quasi-split case
 
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Abstract: In this talk we consider a linear algebraic object called a tridiagonal pair. This object originated in algebraic graph theory, and has connections to orthogonal polynomials and representation theory. In our discussion we aim at a general mathematical audience; no prior experience with the above topics is assumed. Our main results are joint work with Tatsuro Ito and Kazumasa Nomura.
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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.
 
 
The concept of a tridiagonal pair is best explained by starting with a special case called a Leonard pair. Let '''F''' denote a field, and let ''V'' denote a vector space over '''F''' with finite positive dimension. By a Leonard pair on ''V'' we mean a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following two conditions:
 
#There exists a basis for ''V'' with respect to which the matrix representing ''A'' is diagonal and the matrix representing ''A''<sup>∗</sup> is irreducible tridiagonal;
 
#There exists a basis for ''V'' with respect to which the matrix representing ''A''<sup>∗</sup> is diagonal and the matrix representing ''A'' is irreducible tridiagonal.
 
 
 
We recall that a tridiagonal matrix is irreducible whenever each entry on the superdiagonal is nonzero and each entry on the subdiagonal is nonzero. The name "Leonard pair" is motivated by a connection to a 1982 theorem of the combinatorialist Doug Leonard involving the ''q''-Racah and related polynomials of the Askey scheme. Leonard’s theorem was heavily influenced by the work of Eiichi Bannai and Tatsuro Ito on ''P''- and ''Q''- polynomial association schemes, and the work of Richard Askey on orthogonal polynomials. These works in turn were influenced by the work of Philippe Delsarte on coding theory, dating from around 1973.
 
 
 
The central result about Leonard pairs is that they are in bijection with the orthogonal polynomials that make up the terminating branch of the Askey scheme. This branch consists of the ''q''-Racah, ''q''-Hahn, dual ''q''-Hahn, ''q''-Krawtchouk, dual ''q''-Krawtchouk, quantum ''q''-Krawtchouk, affine ''q''-Krawtchouk, Racah, Hahn, dual-Hahn, Krawtchouk, and the Bannai/Ito polynomials. The bijection makes it possible to develop a uniform theory of these polynomials starting from the Leonard pair axiom, and we have done this over the past several years.
 
 
 
A tridiagonal pair is a generalization of a Leonard pair and defined as follows. Let ''V'' denote a vector space over '''F''' with finite positive dimension. A ''tridiagonal pair on V'' is a pair of linear transformations ''A''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and ''A''<sup>∗</sup>&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that satisfy the following four conditions:
 
<ol>
 
<li>Each of ''A'', ''A''<sup>∗</sup> is diagonalizable on ''V'';
 
<li>There exists an ordering {''V<sub>i</sub>''}<sub>''i''=0,...,''d''</sub> of the eigenspaces of ''A'' such that
 
::''A''<sup>*</sup>''V<sub>i</sub>''&nbsp;⊆&nbsp;''V''<sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V<sub>i</sub>''&nbsp;+&nbsp;''V''<sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d''),
 
where ''V''<sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sub>''d''+1</sub>&nbsp;=&nbsp;0;
 
<li>There exists an ordering {''V<sub>i</sub>''<sup>*</sup>}<sub>''i''=0,...,&delta;</sub> of the eigenspaces of ''A''<sup>*</sup> such that
 
::''AV<sub>i</sub>''<sup>*</sup>&nbsp;⊆&nbsp;''V''<sup>*</sup><sub>''i''&minus;1</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''</sub>&nbsp;+&nbsp;''V''<sup>*</sup><sub>''i''+1</sub>&nbsp;&nbsp;(0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;&delta;),
 
where ''V''<sup>*</sup><sub>&minus;1</sub>&nbsp;=&nbsp;0, ''V''<sup>*</sup><sub>''d''+1</sub>&nbsp;=&nbsp;0;
 
<li>There is no subspace ''W''&nbsp;⊆&nbsp;''V'' such that ''AW''&nbsp;⊆&nbsp;W, ''A''<sup>*</sup>''W''&nbsp;⊆&nbsp;''W'', ''W''&nbsp;&ne;&nbsp;0, ''W''&nbsp;&ne;&nbsp;''V''.
 
</ol>
 
It turns out that ''d''&nbsp;=&nbsp;δ and this common value is called the diameter of the pair.
 
A Leonard pair is the same thing as a tridiagonal pair for which the eigenspaces
 
''V'' and ''V''<sup>∗</sup> all have dimension 1.
 
 
 
Tridiagonal pairs arise naturally in the theory of ''P''- and ''Q''-polynomial association schemes, in connection with irreducible modules for the subconstituent algebra. They also appear in recent work of Pascal Baseilhac on the Ising model and related structures in statistical mechanics.
 
 
 
In this talk we will summarize the basic facts about a tridiagonal pair, describing
 
features such as the eigenvalues, dual eigenvalues, shape, tridiagonal relations,
 
split decomposition, and parameter array. We will then focus on a special case
 
said to be sharp and defined as follows. Referring to the tridiagonal pair ''A'', ''A''<sup>∗</sup> in the above definition, it turns out that for 0&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''d'' the dimensions of ''V<sub>i</sub>'', ''V''<sub>''d''&minus;''i''</sub>, ''V''<sup>*</sup><sub>''i''</sub>, ''V''<sup>*</sup><sub>''d''&minus;''i''</sub> coincide; the pair ''A'', ''A''<sup>∗</sup> is called ''sharp'' whenever ''V''<sub>0</sub> has dimension 1. It is known that if '''F''' is algebraically closed then ''A'', ''A''<sup>∗</sup> is sharp.
 
 
 
In our main result we classify the sharp tridiagonal pairs up to isomorphism.
 
 
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Revision as of 21:19, 10 October 2012

September 13

Nigel Boston (UW–Madison)
Title: Non-abelian Cohen–Lenstra heuristics

Abstract: In 1983, Cohen and Lenstra observed that the frequency with which a given abelian p-group A (p odd) arises as the p-class group of an imaginary quadratic field K is apparently proportional to 1/|Aut(A)|. The group A is isomorphic to the Galois group of the maximal unramified abelian p-extension of K. In work with Michael Bush and Farshid Hajir, I generalized this to non-abelian unramified p-extensions of imaginary quadratic fields. I shall recall all the above and describe a further generalization to non-abelian unramified p-extensions of H-extensions of Q, for any p, H, where p does not divide the order of H.


September 20

Simon Marshall (Northwestern)
Title: Multiplicities of automorphic forms on GL2

Abstract: I will discuss some ideas related to the theory of p-adically completed cohomology developed by Frank Calegari and Matthew Emerton. If F is a number field which is not totally real, I will use these ideas to prove a strong upper bound for the dimension of the space of cohomological automorphic forms on GL2 over F which have fixed level and growing weight.


September 27

Jordan Ellenberg (UW–Madison)
Title: Topology of Hurwitz spaces and Cohen-Lenstra conjectures over function fields

Abstract: We will discuss recent progress, joint with Akshay Venkatesh and Craig Westerland, towards the Cohen–Lenstra conjecture over the function field Fq(t). There are two key novelties, one topological and one arithmetic. The first is a homotopy-theoretic description of the "moduli space of G-covers with infinitely many branch points." The second is a description of the stable components of Hurwitz space over Fq, as a module for Gal(Fq/Fq). At least half the talk will be devoted to explaining why these objects are relevant to a very down-to-earth question like Cohen–Lenstra. If time permits, I'll explain what this has to do with the conjectures Nigel spoke about two weeks ago, and a bit about what Daniel is up to.


October 4

Sean Rostami (Madison)
Title: Centers of Hecke algebras

Abstract: The classification and construction of smooth representations of algebraic groups (over non-archimedean local fields) depends heavily on certain function algebras called Hecke algebras. The centers of such algebras are particularly important for classification theorems, and also turn out to be the home of some trace functions that appear in the Hasse–Weil zeta function of a Shimura variety. The Bernstein isomorphism is an explicit identification of the center of an Iwahori–Hecke algebra. I talk about all these things, and outline a satisfying direct proof of the Bernstein isomorphism (the theorem is old, the proof is new).


October 11

Tonghai Yang (Madison)
Title: Quaternions and Kudla's matching principle

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).


October 18

Rachel Davis (Madison)
Title: On the images of metabelian Galois representations associated to elliptic curves

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.


October 25

tbd (tbd)
Title: tba

Abstract: tba


November 1

Lei Zhang (Boston College)
Title: Tensor product L-functions of classical groups of Hermitian type: quasi-split case

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.


March 29

David P. Roberts (U. Minnesota Morris)
Title: Lightly ramified number fields with Galois group S.M12.A

Abstract: Two of the most important invariants of an irreducible polynomial f(x) ∈ Z[x ] are its Galois group G and its field discriminant D. The inverse Galois problem asks one to find a polynomial f(x) having any prescribed Galois group G. Refinements of this problem ask for D to be small in various senses, for example of the form ± pa for the smallest possible prime p.

The talk will discuss this problem in general, with a focus on the technique of specializing three-point covers for solving instances of it. Then it will pursue the cases of the Mathieu group M12, its automorphism group M12.2, its double cover 2.M12, and the combined extension 2.M12.2. Among the polynomials found is

f(x) = x48 + 2 e3 x42 + 69 e5 x36 + 868 e7 x30 − 4174 e7 x26 + 11287 e9 x24
− 4174 e10 x20 + 5340 e12 x18 + 131481 e12 x14 +17599 e14 x12 + 530098 e14 x8
+ 3910 e16 x6 + 4355569 e14 x4 + 20870 e16 x2 + 729 e18,

with e = 11. This polynomial has Galois group G = 2.M12.2 and field discriminant 1188. It makes M12 the first of the twenty-six sporadic simple groups Γ known to have a polynomial with Galois group G involving Γ and field discriminant D the power of a single prime dividing |Γ |.



April 12

Chenyan Wu (Minnesota)
Title: Rallis inner product formula for theta lifts from metaplectic groups to orthogonal groups

Abstract: Let π be a genuine cuspidal representation of the metaplectic group of rank n. We consider the theta lifts to the orthogonal group associated to a quadratic space of dimension 2n + 1. We show a case of a regularized Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands L-function of π twisted by a genuine character. This enables us to demonstrate the relation between non-vanishing of theta lifts and the non-vanishing of central L-values. We prove also a case of regularized Siegel–Weil formula which is missing in the literature, as it forms the basis of our proof of the Rallis inner product formula.



April 16 (special day: Monday, special time: 3:30pm–4:30pm, special place: VV B139)

Hourong Qin (Nanjing U., China)
Title: CM elliptic curves and quadratic polynomials representing primes

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 ap = r for some prime p. We show that there are infinitely many primes p such that ap = r for some fixed integer r if and only if a quadratic polynomial represents infinitely many primes p.


April 19

Robert Guralnick (U. Southern California)
Title: A variant of Burnside and Galois representations which are automorphic

Abstract: Wiles, Taylor, Harris and others used the notion of a big 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—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).


April 26

Frank Thorne (U. South Carolina)
Title: Secondary terms in counting functions for cubic fields

Abstract: We will discuss our proof of secondary terms of order X5/6 in the Davenport–Heilbronn 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.


May 3

Alina Cojocaru (U. Illinois at Chicago)
Title: Frobenius fields for elliptic curves

Abstract: Let E be an elliptic curve defined over Q. For a prime p of good reduction for E, let πp be the p-Weil root of E and Qp) the associated imaginary quadratic field generated by πp. In 1976, Serge Lang and Hale Trotter formulated a conjectural asymptotic formula for the number of primes p < x for which Qp) 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.


May 10

Samit Dasgupta (UC Santa Cruz)
Title: The p-adic L-functions of evil Eisenstein series

Abstract: Let f be a newform of weight k+2 on Γ1(N), and let p ∤ N be a prime. For each root α of the Hecke polynomial of f at p, there is a corresponding p-stabilization fα on Γ1(N) ∩ Γ0(p) with Up-eigenvalue equal to α. The construction of p-adic L-functions associated to such forms fα has been much studied. The non-critical case (when ordp(α) < k+1) was handled in the 1970s via interpolation of the classical L-function in work of Mazur, Swinnerton-Dyer, Manin, Visik, and Amice–Vélu. 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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Organizer contact information

Robert Harron

Zev Klagsbrun

Sean Rostami


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