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= Number Theory / Representation Theory Seminar, University of Wisconsin - Madison =
Return to [https://www.math.wisc.edu/wiki/index.php/NTS ]




*'''When:''' Thursdays, 2:30 PM – 3:30 PM
== Jan 25 ==
*'''Where:''' Van Vleck B113
*Please join the [https://mailhost.math.wisc.edu/mailman/listinfo/nts NT/RT mailing list:] (you must be on a math department computer to use this link).


There is also an accompanying [https://www.math.wisc.edu/wiki/index.php/NTSGrad_Spring_2018 graduate-level seminar], which meets on Tuesdays.<br>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Ali Zaman '''
|-
| bgcolor="#BCD2EE"  align="center" | A log-free zero density estimate for Rankin-Selberg $L$-functions and applications
|-
| bgcolor="#BCD2EE"  | Abstract:We discuss a log-free zero density estimate for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given set of cusp forms and $\pi_0$ is a fixed cusp form.  This estimate is unconditional in many cases of interest, and holds in full generality assuming an average form of the generalized Ramanujan conjecture. There are several applications of this density estimate related to the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and nontrivial bounds for torsion in class groups of number fields assuming the existence of a Siegel zero. We will highlight the latter two topics. This represents joint work with Jesse Thorner. 
|}                                                                       
</center>
 
<br>
 
== Feb 1 ==
 
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yunqing Tang'''
|-
| bgcolor="#BCD2EE"  align="center" | Exceptional splitting of reductions of abelian surfaces with real multiplication
|-
| bgcolor="#BCD2EE"  | Abstract: Zywina showed that after passing to a suitable field extension, every abelian surface $A$ with real multiplication over some number field has geometrically simple reduction modulo $\frak{p}$ for a density one set of primes $\frak{p}$. One may ask whether its complement, the density zero set of primes $\frak{p}$ such that the reduction of $A$ modulo $\frak{p}$ is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod $\frak{p}$ isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.
 
|}                                                                       
</center>
 
 
 
== Feb 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%" | '''Roman Fedorov'''
|-
| bgcolor="#BCD2EE"  align="center" | A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic
|-
| bgcolor="#BCD2EE"  | Abstract: Let G be a reductive group scheme over a regular local ring R. An old conjecture of Grothendieck and Serre predicts that such a principal bundle is trivial, if it is trivial over the fraction field of R. The conjecture has recently been proved in the "geometric" case, that is, when R contains a field. In the remaining case, the difficulty comes from the fact, that the situation is more rigid, so that a certain general position argument does not go through. I will discuss this difficulty and a way to circumvent it to obtain some partial results.
 
|}                                                                       
</center>
 
== Feb 13==
 
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Calegari'''
|-
| bgcolor="#BCD2EE"  align="center" | Recent Progress in Modularity
|-
| bgcolor="#BCD2EE"  | Abstract:  We survey some recent work in modularity lifting, and also describe some applications of these results. This will be based partly on joint work with Allen, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, and also on joint work with Boxer, Gee, and Pilloni.
 
|}                                                                       
</center>
 
== Feb 15 ==
 
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Junho Peter Whang'''
|-
| bgcolor="#BCD2EE"  align="center" | Integral points and curves on moduli of local systems
|-
| bgcolor="#BCD2EE"  | Abstract: We consider the Diophantine geometry of moduli spaces for 
special linear rank two local systems on surfaces with fixed boundary 
traces. After motivating their Diophantine study, we establish a 
structure theorem for their integral points via mapping class group 
descent, generalizing classical work of Markoff (1880). We also obtain 
Diophantine results for algebraic curves in these moduli spaces, 
including effective finiteness of imaginary quadratic integral points 
for non-special curves.
 
|}                                                                       
</center>
 
== Feb 22 ==
 
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifan Yang'''
|-
| bgcolor="#BCD2EE"  align="center" | Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus
|-
| bgcolor="#BCD2EE"  | Abstract: In this talk we consider the rational torsion
subgroup of the generalized Jacobian of the modular
curve X_0(N) with respect to a reduced divisor given
by the sum of all cusps. When N=p is a prime, we find
that the rational torsion subgroup is always cyclic
of order 2 (while that of the usual Jacobian of X_0(p)
grows linearly as p tends to infinity, according to a
well-known result of Mazur). Subject to some unproven
conjecture about the rational torsions of the Jacobian
of X_0(p^n), we also determine the structure of the
rational torsion subgroup of the generalized Jacobian
of X_0(p^n). This is a joint work with Takao Yamazaki.
 
|}                                                                       
</center>
<br>
 
== March 22 ==


<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Fang-Ting Tu'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: Supercongrence for Rigid Hypergeometric Calabi-Yau Threefolds
|-
| bgcolor="#BCD2EE"  | Abstract:
This is a joint work with Ling Long, Noriko Yui, and Wadim Zudilin. We establish the supercongruences for the rigid hypergeometric Calabi-Yau threefolds over rational numbers. These supercongruences were conjectured by Rodriguez-Villeagas in 2003. In this work, we use two different approaches. The first method is based on Dwork's p-adic unit root theory, and the other is based on the theory of hypergeometric motives and hypergeometric functions over finite fields. In this talk, I will introduce the first method, which allows us to obtain the supercongruences for ordinary primes.




= Spring 2018 Semester =
|}                                                                       
</center>
<br>
== April 12 ==


<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Junehyuk Jung'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: Quantum Unique Ergodicity and the number of nodal domains of automorphic forms
|-
| bgcolor="#BCD2EE"  |
Abstract: It has been known for decades that on a flat torus or on a sphere, there exist sequences of eigenfunctions having a bounded number of nodal domains. In contrast, for a manifold with chaotic geodesic flow, the number of nodal domains of eigenfunctions is expected to grow with the eigenvalue. In this talk, I will explain how one can prove that this is indeed true for the surfaces where the Laplacian is quantum uniquely ergodic, under certain symmetry assumptions. As an application, we prove that the number of nodal domains of Maass-Hecke eigenforms on a compact arithmetic triangles tends to $+\infty$ as the eigenvalue grows. I am going to also discuss the nodal domains of automorphic forms on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$. Under a minor assumption, I will give a quick proof that the real part of weight $k\neq 0$ automorphic form has only two nodal domains. This result captures the fact that a 3-manifold with Sasaki metric never admits a chaotic geodesic flow. This talk is based on joint works with S. Zelditch and S. Jang.


{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"
|}                                                                       
</center>
<br>
 
== April 19 ==
 
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hang Xue (Arizona)'''
|-
|-
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''
| bgcolor="#BCD2EE" align="center" | Title: Arithmetic theta lifts and the arithmetic Gan--Gross--Prasad conjecture.
| bgcolor="#F0A0A0" width="300" align="center"|'''Speaker''' (click for homepage)
| bgcolor="#BCD2EE" width="300" align="center"|'''Title''' (click for abstract)
|-
|-
| bgcolor="#E0E0E0" align="center" | Jan  25
| bgcolor="#BCD2EE"  |  
| bgcolor="#F0B0B0" align="center" |[https://profiles.stanford.edu/asif-zaman/ Asif Ali Zaman (Stanford)] 
Abstract: I will explain the arithmetic analogue of the Gan--Gross--Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_1  " A log-free zero density estimate for Rankin-Selberg $L$-functions and applications"]
|-
| bgcolor="#E0E0E0" align="center" | Feb  1
| bgcolor="#F0B0B0" align="center" | [http://web.math.princeton.edu/~yunqingt/ Yunqing Tang (Princeton)] 
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_1  "Exceptional splitting of reductions of abelian surfaces with real multiplication"]
|-
| bgcolor="#E0E0E0" align="center" | Feb  8
| bgcolor="#F0B0B0" align="center" | [http://www.mathematics.pitt.edu/person/roman-fedorov Roman Fedorov (University of Pittsburgh)]
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_1 " A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic"]
|-
| bgcolor="#E0E0E0" align="center" | Feb  13 (note special day!)
| bgcolor="#F0B0B0" align="center" | [http://math.uchicago.edu/~fcale/ Frank Calegari (U. Chicago)]
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_13 "Recent Progress in Modularity"]
|-
| bgcolor="#E0E0E0" align="center" | Feb  15
| bgcolor="#F0B0B0" align="center" |  [https://web.math.princeton.edu/~jwhang/ Junho Peter Whang (Princeton)]
| bgcolor="#BCE2FE"|[https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_15  " Integral points and curves on moduli of local systems"]
|-
| bgcolor="#E0E0E0" align="center" | Feb 22
| bgcolor="#F0B0B0" align="center" | Yifan Yang
| bgcolor="#BCE2FE"|  [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_22  "Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus"] 
|-
| bgcolor="#E0E0E0" align="center" | March 1
| bgcolor="#F0B0B0" align="center" |[http://www.math.wisc.edu/~ellenber/ Jordan Ellenberg (UW Madison)]
| bgcolor="#BCE2FE"|  [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#March_1 "Additive number theory from the algebro-geometric point of view"]
|-
| bgcolor="#E0E0E0" align="center" | March 8
| bgcolor="#F0B0B0" align="center" |
| bgcolor="#BCE2FE"|
|-
| bgcolor="#E0E0E0" align="center" | March 15
| bgcolor="#F0B0B0" align="center" |
| bgcolor="#BCE2FE"|
|-
| bgcolor="#E0E0E0" align="center" | March 22
| bgcolor="#F0B0B0" align="center" | Fang-Ting Tu
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#Feb_15  " Supercongrence for Rigid Hypergeometric Calabi-Yau Threefolds"]
|-
| bgcolor="#E0E0E0" align="center" | March 29
| bgcolor="#F0B0B0" align="center" |
| bgcolor="#BCE2FE"|
|-
| bgcolor="#E0E0E0" align="center" | April 5
| bgcolor="#F0B0B0" align="center" | Special doubleheader!  [http://www.lolathompson.com/ Lola Thompson (Oberlin)] and [https://sites.google.com/site/dmcreyn/home Ben McReynolds (Purdue)], 2:30-4:30
| bgcolor="#BCE2FE"|  "Counting and effective rigidity in algebra and geometry"
|-
| bgcolor="#E0E0E0" align="center" | April 12
| bgcolor="#F0B0B0" align="center" | Junehyuk Jung (Texas A&M)
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#April_12  "Quantum Unique Ergodicity and the number of nodal domains of automorphic forms "]
|-
| bgcolor="#E0E0E0" align="center" | April 19 
| bgcolor="#F0B0B0" align="center" | [http://math.arizona.edu/~xuehang/  Hang Xue (Arizona)]
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#April_12  "Arithmetic theta lifts and the arithmetic Gan--Gross--Prasad conjecture. "]
|-
| bgcolor="#E0E0E0" align="center" | April 26
| bgcolor="#F0B0B0" align="center" |
| bgcolor="#BCE2FE"|
|-
| bgcolor="#E0E0E0" align="center" | May 3
| bgcolor="#F0B0B0" align="center" | [http://www.dms.umontreal.ca/~mlalin/ Matilde Lalin (Université de Montréal)]
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#May_3  "The mean value of cubic $L$-functions over function fields. "]
|-
| bgcolor="#E0E0E0" align="center" | May 10 
| bgcolor="#F0B0B0" align="center" | [http://www.math.harvard.edu/~hpasten/ Hector Pasten (Harvard)]
| bgcolor="#BCE2FE"| [https://www.math.wisc.edu/wiki/index.php/NTS_ABSTRACTSpring2018#May_10  "Shimura curves and estimates for abc triples. "]
|-
| bgcolor="#E0E0E0" align="center" | May 17
| bgcolor="#F0B0B0" align="center" |
| bgcolor="#BCE2FE"|
|-
| bgcolor="#E0E0E0" align="center" | May 24
| bgcolor="#F0B0B0" align="center" |
| bgcolor="#BCE2FE"|


|}
|}                                                                      
</center>
</center>
<br>
== May 3 ==
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Matilde Lalin (Université de Montréal)'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: The mean value of cubic $L$-functions over function fields.
|-
| bgcolor="#BCD2EE"  |
Abstract: We will start by exploring the problem of finding moments for  Dirichlet $L$-functions, including the first main results and the standard conjectures. We will then discuss the problem for function fields. We will then present  a result about the first moment of $L$-functions associated to cubic characters over  $\F_q(t)$, when $q\equiv 1 \bmod{3}$. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. This is joint work with C. David and A. Florea.


|}                                                                       
</center>
<br>
<br>


= Organizer contact information =
== May 10 ==


Naser Talebizadeh Sardari [http://www.math.wisc.edu/~ntalebiz/]
<center>
----
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
Return to the [[Algebra|Algebra Group Page]]
|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hector Pasten (Harvard University)'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: Shimura curves and estimates for abc triples.
|-
| bgcolor="#BCD2EE"  |
Abstract: I will explain a new connection between modular forms and the abc conjecture. In this approach, one considers maps to a given elliptic curve coming from various Shimura curves, which gives a way to obtain unconditional results towards the abc conjecture starting from good estimates for the variation of the degree of these maps. The approach to control this variation of degrees involves a number of tools, such as Arakelov geometry, automorphic forms, and analytic number theory. The final result is an unconditional estimate that lies beyond the existing techniques in the context of the abc conjecture, such as linear forms in logarithms.
|}                                                                       
</center>
<br>

Revision as of 12:43, 16 January 2019

Return to [1]


Jan 25

Asif Ali Zaman
A log-free zero density estimate for Rankin-Selberg $L$-functions and applications
Abstract:We discuss a log-free zero density estimate for Rankin-Selberg $L$-functions of the form $L(s,\pi\times\pi_0)$, where $\pi$ varies in a given set of cusp forms and $\pi_0$ is a fixed cusp form. This estimate is unconditional in many cases of interest, and holds in full generality assuming an average form of the generalized Ramanujan conjecture. There are several applications of this density estimate related to the rarity of Landau-Siegel zeros of Rankin-Selberg $L$-functions, the Chebotarev density theorem, and nontrivial bounds for torsion in class groups of number fields assuming the existence of a Siegel zero. We will highlight the latter two topics. This represents joint work with Jesse Thorner.


Feb 1

Yunqing Tang
Exceptional splitting of reductions of abelian surfaces with real multiplication
Abstract: Zywina showed that after passing to a suitable field extension, every abelian surface $A$ with real multiplication over some number field has geometrically simple reduction modulo $\frak{p}$ for a density one set of primes $\frak{p}$. One may ask whether its complement, the density zero set of primes $\frak{p}$ such that the reduction of $A$ modulo $\frak{p}$ is not geometrically simple, is infinite. Such question is analogous to the study of exceptional mod $\frak{p}$ isogeny between two elliptic curves in the recent work of Charles. In this talk, I will show that abelian surfaces over number fields with real multiplication have infinitely many non-geometrically-simple reductions. This is joint work with Ananth Shankar.


Feb 8

Roman Fedorov
A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic
Abstract: Let G be a reductive group scheme over a regular local ring R. An old conjecture of Grothendieck and Serre predicts that such a principal bundle is trivial, if it is trivial over the fraction field of R. The conjecture has recently been proved in the "geometric" case, that is, when R contains a field. In the remaining case, the difficulty comes from the fact, that the situation is more rigid, so that a certain general position argument does not go through. I will discuss this difficulty and a way to circumvent it to obtain some partial results.

Feb 13

Frank Calegari
Recent Progress in Modularity
Abstract: We survey some recent work in modularity lifting, and also describe some applications of these results. This will be based partly on joint work with Allen, Caraiani, Gee, Helm, Le Hung, Newton, Scholze, Taylor, and Thorne, and also on joint work with Boxer, Gee, and Pilloni.

Feb 15

Junho Peter Whang
Integral points and curves on moduli of local systems
Abstract: We consider the Diophantine geometry of moduli spaces for

special linear rank two local systems on surfaces with fixed boundary traces. After motivating their Diophantine study, we establish a structure theorem for their integral points via mapping class group descent, generalizing classical work of Markoff (1880). We also obtain Diophantine results for algebraic curves in these moduli spaces, including effective finiteness of imaginary quadratic integral points for non-special curves.

Feb 22

Yifan Yang
Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus
Abstract: In this talk we consider the rational torsion

subgroup of the generalized Jacobian of the modular curve X_0(N) with respect to a reduced divisor given by the sum of all cusps. When N=p is a prime, we find that the rational torsion subgroup is always cyclic of order 2 (while that of the usual Jacobian of X_0(p) grows linearly as p tends to infinity, according to a well-known result of Mazur). Subject to some unproven conjecture about the rational torsions of the Jacobian of X_0(p^n), we also determine the structure of the rational torsion subgroup of the generalized Jacobian of X_0(p^n). This is a joint work with Takao Yamazaki.


March 22

Fang-Ting Tu
Title: Supercongrence for Rigid Hypergeometric Calabi-Yau Threefolds
Abstract:

This is a joint work with Ling Long, Noriko Yui, and Wadim Zudilin. We establish the supercongruences for the rigid hypergeometric Calabi-Yau threefolds over rational numbers. These supercongruences were conjectured by Rodriguez-Villeagas in 2003. In this work, we use two different approaches. The first method is based on Dwork's p-adic unit root theory, and the other is based on the theory of hypergeometric motives and hypergeometric functions over finite fields. In this talk, I will introduce the first method, which allows us to obtain the supercongruences for ordinary primes.



April 12

Junehyuk Jung
Title: Quantum Unique Ergodicity and the number of nodal domains of automorphic forms

Abstract: It has been known for decades that on a flat torus or on a sphere, there exist sequences of eigenfunctions having a bounded number of nodal domains. In contrast, for a manifold with chaotic geodesic flow, the number of nodal domains of eigenfunctions is expected to grow with the eigenvalue. In this talk, I will explain how one can prove that this is indeed true for the surfaces where the Laplacian is quantum uniquely ergodic, under certain symmetry assumptions. As an application, we prove that the number of nodal domains of Maass-Hecke eigenforms on a compact arithmetic triangles tends to $+\infty$ as the eigenvalue grows. I am going to also discuss the nodal domains of automorphic forms on $SL_2(\mathbb{Z})\backslash SL_2(\mathbb{R})$. Under a minor assumption, I will give a quick proof that the real part of weight $k\neq 0$ automorphic form has only two nodal domains. This result captures the fact that a 3-manifold with Sasaki metric never admits a chaotic geodesic flow. This talk is based on joint works with S. Zelditch and S. Jang.


April 19

Hang Xue (Arizona)
Title: Arithmetic theta lifts and the arithmetic Gan--Gross--Prasad conjecture.

Abstract: I will explain the arithmetic analogue of the Gan--Gross--Prasad conjecture for unitary groups. I will also explain how to use arithmetic theta lift to prove certain endoscopic cases of it.


May 3

Matilde Lalin (Université de Montréal)
Title: The mean value of cubic $L$-functions over function fields.

Abstract: We will start by exploring the problem of finding moments for Dirichlet $L$-functions, including the first main results and the standard conjectures. We will then discuss the problem for function fields. We will then present a result about the first moment of $L$-functions associated to cubic characters over $\F_q(t)$, when $q\equiv 1 \bmod{3}$. The case of number fields was considered in previous work, but never for the full family of cubic twists over a field containing the third roots of unity. This is joint work with C. David and A. Florea.


May 10

Hector Pasten (Harvard University)
Title: Shimura curves and estimates for abc triples.

Abstract: I will explain a new connection between modular forms and the abc conjecture. In this approach, one considers maps to a given elliptic curve coming from various Shimura curves, which gives a way to obtain unconditional results towards the abc conjecture starting from good estimates for the variation of the degree of these maps. The approach to control this variation of degrees involves a number of tools, such as Arakelov geometry, automorphic forms, and analytic number theory. The final result is an unconditional estimate that lies beyond the existing techniques in the context of the abc conjecture, such as linear forms in logarithms.