Scroll down for abstracts for the most recent seminars
Thurs. Sep. 13----Frank Thorne (UW-Madison)
Title: Maier Matrices beyond $\mathbb{Z}$
Abstract: A "Maier matrix" is a combinatorial device used to prove the existence of irregular and unusual behavior in the distribution of the primes and other arithmetic sequences. In the first part of the talk, I will give an overview of the method and describe several interesting results which it can be used to prove. In particular, I will describe the "uncertainty principle" of Granville and Soundarajan, which proves the existence of irregularities in a very general class of sequences. In the second part of the talk I will describe my own work extending the method to $\mathbb{F}_q[t]$ and to certain number fields, where I was able to prove many similar results.
Thurs. Sep. 20----Tonghai Yang (UW-Madison)
Title: Central L-values of Hecke characters
Abstract: Algebraic Hecke characters are associated to Abelian varieties (ellptic curves) with complex multiplications or more generally CM motives. Their central L-values have thus arithmetic meanings. In this talk, I will describe an Asymptotic formula for the average of Hecke L-values over a `sub'family of Hecke characters. We will also prove some quantative non-vanishing theorems for these L-values. This is joint work with Riad Masri.
Thurs. Sep. 27----Matt Bainbridge (Univ of Chicago)
Title: Teichmuller curves on Hilbert modular surfaces
Abstract: A Teichmuller curve is an algebraic curve in the moduli space of curves which is isometrically immersed with respect to certain metrics. There are many surprising connecions between Teichmuller curves and number theory. In this talk, I'll discuss some of these connections, focusing on a series of examples, discovered recently by McMullen, of Teichmuller curves which are naturally embedded in Hilbert modular surfaces. The study of these examples involves Hilbert modular forms and compactifications of Hilbert modular surfaces arising from the Deligne-Mumford compactification of moduli space.
Thurs. Oct. 4----Amanda Folsom (UW-Madison)
Title: Half-integral weight Maass form correspondences and vector valued forms
Abstract: Recent celebrated works of Zwegers and Bringmann-Ono have placed the mock $\Theta$-functions and their generalizations in the context of weight $\frac{1}{2}$ harmonic weak Maass forms. In light of this, one expects similar correspondences to hold between other spaces of half-integral weight Maass forms, however missing are natural candidates to serve as analogues to the mock $\Theta$-functions. In separate works with Bringmann-Ono and Bruinier-Bringmann-Ono, we make such correspondences precise by constructing half-integral weight vector valued harmonic weak Maass forms on the full modular group $\SL_2(\mathbb Z)$ whose transformation properties are dictated by the Weil representation arising from elementary theta series. We show that these vector valued Maass forms give rise to certain families of hypergeometric series and also Borcherds products. In both cases we establish an isomorphism between spaces of half-integral weight Maass forms and classical spaces of half-integral weight modular forms.
Thurs. Oct. 11----Kartik Prasanna (Maryland)
Title: Elliptic curves, quadratic twists and p-(in)divisibility of L-values
Abstract: Let E be an elliptic curve over the rationals and L(E,s) the L-function associated to E. A famous theorem of Waldspurger states that there exists a quadratic discriminant d such that the central value L(E_d, 1) is nonzero, where E_d is the d'th quadratic twist of E. This theorem has many applications, most notably to the Birch and Swinnerton-Dyer conjecture in the case of rank 1 elliptic curves. I will explain a conjectural "mod p" version of Waldspurger's theorem and some related results that are obtained by studying the p-adic properties of the Shimura correspondence.
Thurs. Nov. 1----Toby Gee (Northwestern)
Title: Generalisations of the weight part of Serre's conjecture
Abstract: I will give an overview of recent work on formulating generalisations of the weight part of Serre's conjecture to other reductive groups. I will also discuss potential applications.
Thurs. Nov. 6----Guangwu Xu (UW-Milwaukee)
Title: Efficient computation of Weil/Tate pairing
Abstract: Over the past decade, we have seen several remarkable applications of Weil/Tate pairings in cryptography. For example, they are used for the reduction of discrete logarithm problem for certain class of elliptic curves. They are also basic building blocks for some exciting new cryptographic primitives. In this talk, we shall start by describing some cryptographic constructions using pairings. Then we shall discuss the Weil/Tate pairings on elliptic curves, and Miller's algorithm for computing pairings. In the last part, we shall discuss some refinements of Miller's algorithm. (This talk is based on a work joint with Ian F. Blake and V. Kumar Murty)
Thurs. Dec. 6----Jorge Jiminez Urroz (Universidad Politecnica de Catalunya)
Title: On the Factorization of RSA moduli
Abstract: This is joint work with L. Dieulefait. It is hard to factorize an RSA modulus in the most simple sense: somebody gives us a product of two primes n=pq, (slightly big say), and, in general, we do not know how to find p and q. In 2006 P. Paillier and J. Villar made the conjecture that this problem would still be hard even if we know how to factorize integers n' coprime with n. (They do it with the help of certain oracle).
Even that the conjecture seems completely reasonable, it is subtle how to formulate it and, in fact, in the talk we will build, for any modulus n, certain explicit n' coprime to n which will indeed allow us to factorize n, knowing only very partial information about the factorization of n' itself
Thurs. Dec. 6----Patrick Rault (UW-Madison)
Title: On uniform bounds for rational points on rational curves and thin sets
Abstract: We show that the number of rational points of height less than $B$, on the image of a degree 2 map from $\mathbb P^1$ to $\mathbb P^n$ ($n\geq 1$), under certain conditions, is $O(B)$, where the point is that the implied constant is independent of the choice of the map. This theorem improves on a result of Heath-Brown and Browning Heath-Brown in the case of degree 2 plane curves. Heath-Brown proved that for any $\epsilon>0$ the number of rational points of height less than $B$ on a degree $d$ plane curve is $O_{\epsilon,d}(B^{2/d+\epsilon})$ (the implied constant depends on $\epsilon$ and $d$). Browning and Heath-Brown later proved that this result holds with $\epsilon=0$ for degree 2 curves. It is known that Heath-Brown's theorem is sharp apart from the $\epsilon$, and in fact Ellenberg and Venkatesh have proven that there is some $\delta>0$ for which the counting function for any plane curve of positive genus is $O_d(B^{2/d-\delta})$. It is an open question whether Heath-Brown's Theorem is true with $\epsilon=0$.