NTS ABSTRACTSpring2019: Difference between revisions

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== Jan 25 ==
== Jan 23 ==


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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Asif Ali Zaman '''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yunqing Tang '''
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| bgcolor="#BCD2EE"  align="center" | A log-free zero density estimate for Rankin-Selberg $L$-functions and applications
| bgcolor="#BCD2EE"  align="center" | Reductions of abelian surfaces over global function fields
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| 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.
| bgcolor="#BCD2EE"  | For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.
 
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== Feb 1 ==
== Jan 24 ==
 
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hassan-Mao-Smith--Zhu'''
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| bgcolor="#BCD2EE"  align="center" | The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of  $S^{d-2}\subset S^d$
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| bgcolor="#BCD2EE"  | Abstract: Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj},  $d\geq 3,$ and $q$ and $p$ prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$  point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the  diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.
 
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== Jan 31 ==
 
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Kyle Pratt'''
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| bgcolor="#BCD2EE"  align="center" | Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions
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| bgcolor="#BCD2EE"  | Abstract: I will discuss recent work, joint with Bui, Robles, and Zaharescu, on a moment problem for Dirichlet $L$-functions. By way of motivation I will spend some time discussing the Lindel\"of Hypothesis, and work of Bettin, Chandee, and Radziwi\l\l. The talk will be accessible, as I will give lots of background information and will not dwell on technicalities.
 
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== Feb 7 ==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yunqing Tang'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Shamgar Gurevich'''
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|-
| bgcolor="#BCD2EE"  align="center" | Exceptional splitting of reductions of abelian surfaces with real multiplication
| bgcolor="#BCD2EE"  align="center" | Harmonic Analysis on $GL_n$ over finite fields
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| 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.
| bgcolor="#BCD2EE"  | Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters.
For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}:
$$trace (\rho(g))/dim (\rho),$$
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (TAMU).


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== Feb 14 ==


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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Tonghai Yang'''
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| bgcolor="#BCD2EE"  align="center" | The Lambda invariant and its CM values
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| bgcolor="#BCD2EE"  | Abstract:  The Lambda invariant which parametrizes  elliptic curves with two torsions (X_0(2)) has some interesting properties, some similar to that of the j-invariants, and some  not. For example,  $\lambda(\frac{d+\sqrt d}2)$  is a unit sometime. In this talk, I will briefly describe some of the properties.  This is joint work with Hongbo Yin and Peng Yu.
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== Feb 8 ==
== Feb 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"
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|-
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Roman Fedorov'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Brian Lawrence'''
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|-
| bgcolor="#BCD2EE"  align="center" | A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic
| bgcolor="#BCD2EE"  align="center" | Diophantine problems and a p-adic period map.
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| 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.
| bgcolor="#BCD2EE"  | Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. Joint with Akshay Venkatesh.


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== Feb 13==
== March 7==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Frank Calegari'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Masoud Zargar'''
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| bgcolor="#BCD2EE"  align="center" | Recent Progress in Modularity
| bgcolor="#BCD2EE"  align="center" | Sections of quadrics over the affine line
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| 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.
| bgcolor="#BCD2EE"  | Abstract: Abstract: Suppose we have a quadratic form Q(x) in d\geq 4 variables over F_q[t] and f(t) is a polynomial over F_q. We consider the affine variety X given by the equation Q(x)=f(t) as a family of varieties over the affine line A^1_{F_q}. Given finitely many closed points in distinct fibers of this family, we ask when there exists a section passing through these points. We study this problem using the circle method over F_q((1/t)). Time permitting, I will mention connections to Lubotzky-Phillips-Sarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari


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== Feb 15 ==
== March 14==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Junho Peter Whang'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Elena Mantovan'''
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|-
| bgcolor="#BCD2EE"  align="center" | Integral points and curves on moduli of local systems
| bgcolor="#BCD2EE"  align="center" | p-adic automorphic forms, differential operators and Galois representations
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| bgcolor="#BCD2EE"  | Abstract: We consider the Diophantine geometry of moduli spaces for 
| bgcolor="#BCD2EE"  | A strategy pioneered by Serre and Katz in the 1970s yields a construction of p-adic families of modular forms via the study of Serre's weight-raising differential operator Theta. This construction is a key ingredient in Deligne-Serre's theorem associating Galois representations to modular forms of weight 1, and in the study of the weight part of Serre's conjecture. In this talk I will discuss recent progress towards generalizing this theory to automorphic forms on unitary and symplectic Shimura varieites. In particular, I will introduce certain p-adic analogues of Maass-Shimura weight-raising differential operatorsand  discuss their action on p-adic automorphic forms, and on the associated mod p Galois representations. In contrast with Serre's classical approach where q-expansions play a prominent role, our approach is geometric in nature and is inspired by earlier work of Katz and Gross.
special linear rank two local systems on surfaces with fixed boundary 
This talk is based joint work with Eishenand also with Fintzen--Varma, and with Flander--Ghitza--McAndrew.
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.


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== Feb 22 ==
== March 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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Yifan Yang'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Adebisi Agboola'''
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|-
| bgcolor="#BCD2EE"  align="center" | Rational torsion on the generalized Jacobian of a modular curve with cuspidal modulus
| bgcolor="#BCD2EE"  align="center" |Relative K-groups and rings of integers
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| bgcolor="#BCD2EE"  | Abstract: In this talk we consider the rational torsion
| bgcolor="#BCD2EE"  | Abstract: Suppose that F is a number field and G is a finite group. I shall discuss a conjecture in relative algebraic K-theory (in essence, a conjectural Hasse principle applied to certain relative algebraic K-groups) that implies an affirmative answer to both the inverse Galois problem for F and G and to an analogous problem concerning the Galois module structure of rings of integers in tame extensions of F. It also implies the weak Malle conjecture on counting tame G-extensions of F according to discriminant. The K-theoretic conjecture can be proved in many cases (subject to mild technical conditions), e.g. when G is of odd order, giving a partial analogue of a classical theorem of Shafarevich in this setting. While this approach does not, as yet, resolve any new cases of the inverse Galois problem, it does yield substantial new results concerning both the Galois module structure of rings of integers and the weak Malle conjecture.
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.


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== March 22 ==
== April 4==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Fang-Ting Tu'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Wei-Lun Tsai'''
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| bgcolor="#BCD2EE"  align="center" | Title: Supercongrence for Rigid Hypergeometric Calabi-Yau Threefolds
| bgcolor="#BCD2EE"  align="center" |Hecke L-functions and $\ell$ torsion in class groups
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| bgcolor="#BCD2EE"  | Abstract:  
| bgcolor="#BCD2EE"  | Abstract: The canonical Hecke characters in the sense of Rohrlich form a
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.  
set of algebraic Hecke characters with important arithmetic properties.
In this talk, we will explain how one can prove quantitative
nonvanishing results for the central values of their corresponding
L-functions using methods of an arithmetic statistical flavor. In  
particular, the methods used rely crucially on recent work of Ellenberg,
Pierce, and Wood concerning bounds for $\ell$-torsion in class groups of
number fields. This is joint work with Byoung Du Kim and Riad Masri.
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== April 11==


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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Taylor Mcadam'''
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| bgcolor="#BCD2EE"  align="center" |Almost-prime times in horospherical flows
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| bgcolor="#BCD2EE"  | Abstract: Equidistribution results play an important role in dynamical systems and their applications in number theory.  Often in such applications it is desirable for equidistribution to be effective (i.e. the rate of convergence is known). In this talk I will discuss some of the history of effective equidistribution results in homogeneous dynamics and give an effective result for horospherical flows on the space of lattices. I will then describe an application to studying the distribution of almost-prime times in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture.
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== April 12 ==
== April 18==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Junehyuk Jung'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Ila Varma'''
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|-
| bgcolor="#BCD2EE"  align="center" | Title: Quantum Unique Ergodicity and the number of nodal domains of automorphic forms
| bgcolor="#BCD2EE"  align="center" |Malle's Conjecture for octic $D_4$-fields.
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| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract: We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.
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.
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== April 25==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Michael Bush'''
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| bgcolor="#BCD2EE"  align="center" |Interactions between group theory and number theory
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| bgcolor="#BCD2EE"  | Abstract: I'll survey some of the ways in which group theory has helped us understand extensions of number fields with restricted ramification and why one might care about such things. Some of Nigel's contributions will be highlighted. A good portion of the talk should be accessible to those other than number theorists.
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== April 19 ==
== April 25==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hang Xue (Arizona)'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Rafe Jones'''
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| bgcolor="#BCD2EE"  align="center" | Title: Arithmetic theta lifts and the arithmetic Gan--Gross--Prasad conjecture.
| bgcolor="#BCD2EE"  align="center" |Eventually stable polynomials and arboreal Galois representations
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| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract: Call a polynomial defined over a field K eventually stable if its nth iterate has a uniformly bounded number of irreducible factors (over K) as n grows. I’ll discuss some far-reaching conjectures on eventual stability, and recent work on various special cases. I’ll also describe some natural connections between eventual stability and arboreal Galois representations, which Nigel Boston introduced in the early 2000s.  
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.
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==April 25 NTS==
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Jen Berg'''
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| bgcolor="#BCD2EE"  align="center" |Rational points on conic bundles over elliptic curves with positive rank
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| bgcolor="#BCD2EE"  | Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses the Brauer group, Br X, to preclude the existence of rational points on a variety X. In this talk, we'll explore the arithmetic of conic bundles over elliptic curves of positive rank over a number field k. We'll discuss the insufficiency of the known obstructions to explain the failures of the Hasse principle for such varieties over a number field. We'll further consider questions on the distribution of the rational points of X with respect to the image of X(k) inside of the rational points of the elliptic curve E. In the process, we'll discuss results on a local-to-global principle for torsion points on elliptic curves over Q. This is joint work in progress with Masahiro Nakahara.
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== May 3 ==
== April 25==


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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"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Matilde Lalin (Université de Montréal)'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Judy Walker'''
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| bgcolor="#BCD2EE"  align="center" | Title: The mean value of cubic $L$-functions over function fields.
| bgcolor="#BCD2EE"  align="center" |Derangements of Finite Groups
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| bgcolor="#BCD2EE"  | Abstract: In the early 1990’s, Nigel Boston taught an innovative graduate-level group theory course at the University of Illinois that focused on derangements (fixed-point-free elements) of transitive permutation groups. The course culminated in the writing of a 7-authored paper that appeared in Communications in Algebra in 1993. This paper contained a conjecture that was eventually proven by Fulman and Guralnick, with that result appearing in the Transactions of the American Mathematical Society just last year.
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.
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== May 2==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Melanie Matchett Wood'''
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| bgcolor="#BCD2EE"  align="center" |Unramified extensions of random global fields
|-
| bgcolor="#BCD2EE"  | Abstract: For any finite group Gamma, I will give a "non-abelian-Cohen-Martinet Conjecture," i.e. a conjectural distribution on the "good part" of the Galois group of the maximal unramified extension of a global field K, as K varies over all Galois Gamma extensions of the rationals or rational function field over a finite field.  I will explain the motivation for this conjecture based on what we know about these maximal unramified extensions (very little), and how we prove, in the function field case, as the size of the finite field goes to infinity, that the moments of the Galois groups of these maximal unramified extensions match out conjecture.  This talk covers work in progress with Yuan Liu and David Zureick-Brown
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== May 10 ==
== May 9==


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| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''Hector Pasten (Harvard University)'''
| bgcolor="#F0A0A0" align="center" style="font-size:125%" | '''David Zureick-Brown'''
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| bgcolor="#BCD2EE"  align="center" | Title: Shimura curves and estimates for abc triples.
| bgcolor="#BCD2EE"  align="center" |Arithmetic of stacks
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| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract: I'll discuss several diophantine problems that naturally lead one to study algebraic stacks, and discuss a few results.  
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.
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Latest revision as of 17:23, 3 May 2019

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Jan 23

Yunqing Tang
Reductions of abelian surfaces over global function fields
For a non-isotrivial ordinary abelian surface $A$ over a global function field, under mild assumptions, we prove that there are infinitely many places modulo which $A$ is geometrically isogenous to the product of two elliptic curves. This result can be viewed as a generalization of a theorem of Chai and Oort. This is joint work with Davesh Maulik and Ananth Shankar.


Jan 24

Hassan-Mao-Smith--Zhu
The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$
Abstract: Assume a polynomial-time algorithm for factoring integers, Conjecture~\ref{conj}, $d\geq 3,$ and $q$ and $p$ prime numbers, where $p\leq q^A$ for some $A>0$. We develop a polynomial-time algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d-2}\subset S^{d}$ to a $\mathbb{Z}[1/p]$ point of $S^d$ with the minimum height. We implement our algorithm for $d=3 \text{ and }4$. Based on our numerical results, we formulate a conjecture which can be checked in polynomial-time and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d-2}\subset S^d$.


Jan 31

Kyle Pratt
Breaking the $\frac{1}{2}$-barrier for the twisted second moment of Dirichlet $L$-functions
Abstract: I will discuss recent work, joint with Bui, Robles, and Zaharescu, on a moment problem for Dirichlet $L$-functions. By way of motivation I will spend some time discussing the Lindel\"of Hypothesis, and work of Bettin, Chandee, and Radziwi\l\l. The talk will be accessible, as I will give lots of background information and will not dwell on technicalities.

Feb 7

Shamgar Gurevich
Harmonic Analysis on $GL_n$ over finite fields
Abstract: There are many formulas that express interesting properties of a group G in terms of sums over its characters.

For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: $$trace (\rho(g))/dim (\rho),$$ for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (TAMU).

Feb 14

Tonghai Yang
The Lambda invariant and its CM values
Abstract: The Lambda invariant which parametrizes elliptic curves with two torsions (X_0(2)) has some interesting properties, some similar to that of the j-invariants, and some not. For example, $\lambda(\frac{d+\sqrt d}2)$ is a unit sometime. In this talk, I will briefly describe some of the properties. This is joint work with Hongbo Yin and Peng Yu.

Feb 28

Brian Lawrence
Diophantine problems and a p-adic period map.
Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. Joint with Akshay Venkatesh.

March 7

Masoud Zargar
Sections of quadrics over the affine line
Abstract: Abstract: Suppose we have a quadratic form Q(x) in d\geq 4 variables over F_q[t] and f(t) is a polynomial over F_q. We consider the affine variety X given by the equation Q(x)=f(t) as a family of varieties over the affine line A^1_{F_q}. Given finitely many closed points in distinct fibers of this family, we ask when there exists a section passing through these points. We study this problem using the circle method over F_q((1/t)). Time permitting, I will mention connections to Lubotzky-Phillips-Sarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari

March 14

Elena Mantovan
p-adic automorphic forms, differential operators and Galois representations
A strategy pioneered by Serre and Katz in the 1970s yields a construction of p-adic families of modular forms via the study of Serre's weight-raising differential operator Theta. This construction is a key ingredient in Deligne-Serre's theorem associating Galois representations to modular forms of weight 1, and in the study of the weight part of Serre's conjecture. In this talk I will discuss recent progress towards generalizing this theory to automorphic forms on unitary and symplectic Shimura varieites. In particular, I will introduce certain p-adic analogues of Maass-Shimura weight-raising differential operators, and discuss their action on p-adic automorphic forms, and on the associated mod p Galois representations. In contrast with Serre's classical approach where q-expansions play a prominent role, our approach is geometric in nature and is inspired by earlier work of Katz and Gross.

This talk is based joint work with Eishen, and also with Fintzen--Varma, and with Flander--Ghitza--McAndrew.

March 28

Adebisi Agboola
Relative K-groups and rings of integers
Abstract: Suppose that F is a number field and G is a finite group. I shall discuss a conjecture in relative algebraic K-theory (in essence, a conjectural Hasse principle applied to certain relative algebraic K-groups) that implies an affirmative answer to both the inverse Galois problem for F and G and to an analogous problem concerning the Galois module structure of rings of integers in tame extensions of F. It also implies the weak Malle conjecture on counting tame G-extensions of F according to discriminant. The K-theoretic conjecture can be proved in many cases (subject to mild technical conditions), e.g. when G is of odd order, giving a partial analogue of a classical theorem of Shafarevich in this setting. While this approach does not, as yet, resolve any new cases of the inverse Galois problem, it does yield substantial new results concerning both the Galois module structure of rings of integers and the weak Malle conjecture.

April 4

Wei-Lun Tsai
Hecke L-functions and $\ell$ torsion in class groups
Abstract: The canonical Hecke characters in the sense of Rohrlich form a

set of algebraic Hecke characters with important arithmetic properties. In this talk, we will explain how one can prove quantitative nonvanishing results for the central values of their corresponding L-functions using methods of an arithmetic statistical flavor. In particular, the methods used rely crucially on recent work of Ellenberg, Pierce, and Wood concerning bounds for $\ell$-torsion in class groups of number fields. This is joint work with Byoung Du Kim and Riad Masri.

April 11

Taylor Mcadam
Almost-prime times in horospherical flows
Abstract: Equidistribution results play an important role in dynamical systems and their applications in number theory. Often in such applications it is desirable for equidistribution to be effective (i.e. the rate of convergence is known). In this talk I will discuss some of the history of effective equidistribution results in homogeneous dynamics and give an effective result for horospherical flows on the space of lattices. I will then describe an application to studying the distribution of almost-prime times in horospherical orbits and discuss connections of this work to Sarnak’s Mobius disjointness conjecture.

April 18

Ila Varma
Malle's Conjecture for octic $D_4$-fields.
Abstract: We consider the family of normal octic fields with Galois group $D_4$, ordered by their discriminant. In forthcoming joint work with Arul Shankar, we verify the strong Malle conjecture for this family of number fields, obtaining the order of growth as well as the constant of proportionality. In this talk, we will discuss and review the combination of techniques from analytic number theory and geometry-of-numbers methods used to prove these results.

April 25

Michael Bush
Interactions between group theory and number theory
Abstract: I'll survey some of the ways in which group theory has helped us understand extensions of number fields with restricted ramification and why one might care about such things. Some of Nigel's contributions will be highlighted. A good portion of the talk should be accessible to those other than number theorists.

April 25

Rafe Jones
Eventually stable polynomials and arboreal Galois representations
Abstract: Call a polynomial defined over a field K eventually stable if its nth iterate has a uniformly bounded number of irreducible factors (over K) as n grows. I’ll discuss some far-reaching conjectures on eventual stability, and recent work on various special cases. I’ll also describe some natural connections between eventual stability and arboreal Galois representations, which Nigel Boston introduced in the early 2000s.

April 25 NTS

Jen Berg
Rational points on conic bundles over elliptic curves with positive rank
Abstract: Varieties that fail to have rational points despite having local points for each prime are said to fail the Hasse principle. A systematic tool accounting for these failures is called the Brauer-Manin obstruction, which uses the Brauer group, Br X, to preclude the existence of rational points on a variety X. In this talk, we'll explore the arithmetic of conic bundles over elliptic curves of positive rank over a number field k. We'll discuss the insufficiency of the known obstructions to explain the failures of the Hasse principle for such varieties over a number field. We'll further consider questions on the distribution of the rational points of X with respect to the image of X(k) inside of the rational points of the elliptic curve E. In the process, we'll discuss results on a local-to-global principle for torsion points on elliptic curves over Q. This is joint work in progress with Masahiro Nakahara.

April 25

Judy Walker
Derangements of Finite Groups
Abstract: In the early 1990’s, Nigel Boston taught an innovative graduate-level group theory course at the University of Illinois that focused on derangements (fixed-point-free elements) of transitive permutation groups. The course culminated in the writing of a 7-authored paper that appeared in Communications in Algebra in 1993. This paper contained a conjecture that was eventually proven by Fulman and Guralnick, with that result appearing in the Transactions of the American Mathematical Society just last year.


May 2

Melanie Matchett Wood
Unramified extensions of random global fields
Abstract: For any finite group Gamma, I will give a "non-abelian-Cohen-Martinet Conjecture," i.e. a conjectural distribution on the "good part" of the Galois group of the maximal unramified extension of a global field K, as K varies over all Galois Gamma extensions of the rationals or rational function field over a finite field. I will explain the motivation for this conjecture based on what we know about these maximal unramified extensions (very little), and how we prove, in the function field case, as the size of the finite field goes to infinity, that the moments of the Galois groups of these maximal unramified extensions match out conjecture. This talk covers work in progress with Yuan Liu and David Zureick-Brown

May 9

David Zureick-Brown
Arithmetic of stacks
Abstract: I'll discuss several diophantine problems that naturally lead one to study algebraic stacks, and discuss a few results.