Difference between revisions of "NTS ABSTRACTSpring2019"
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 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 LubotzkyPhillipsSarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari   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 LubotzkyPhillipsSarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari  
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+  == March 14==  
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+  <center>  
+  { style="color:black; fontsize:100%" table border="2" cellpadding="10" width="700" cellspacing="20"  
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+   bgcolor="#F0A0A0" align="center" style="fontsize:125%"  '''Elena Mantovan'''  
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+   bgcolor="#BCD2EE" align="center"  padic automorphic forms, differential operators and Galois representations  
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+   bgcolor="#BCD2EE"  A strategy pioneered by Serre and Katz in the 1970s yields a construction of padic families of modular forms via the study of Serre's weightraising differential operator Theta. This construction is a key ingredient in DeligneSerre'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 padic analogues of MaassShimura weightraising differential operators, and discuss their action on padic automorphic forms, and on the associated mod p Galois representations. In contrast with Serre's classical approach where qexpansions 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 FintzenVarma, and with FlanderGhitzaMcAndrew. 
Revision as of 11:20, 5 March 2019
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Jan 23
Yunqing Tang 
Reductions of abelian surfaces over global function fields 
For a nonisotrivial 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
HassanMaoSmithZhu 
The diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d2}\subset S^d$ 
Abstract: Assume a polynomialtime 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 polynomialtime algorithm in $\log(q)$ that lifts every $\mathbb{Z}/q\mathbb{Z}$ point of $S^{d2}\subset S^{d}$ 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 polynomialtime and gives the optimal bound on the diophantine exponent of the $\mathbb{Z}/q\mathbb{Z}$ points of $S^{d2}\subset S^d$. 
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 Gbiinvariant 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 jinvariants, 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 padic period map. 
Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using padic 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 LubotzkyPhillipsSarnak (LPS) Ramanujan graphs. Joint with Naser T. Sardari
March 14<center>
