Number Theory Seminar University of South Carolina, Fall 2012

Transcription

1 Number Theory Seminar University of South Carolina, Fall 2012 The seminar was organized by Matt Boylan, Michael Filaseta, and Frank Thorne. We warmly welcome outside visitors. Please contact any of the organizers if you are visiting and would like to speak! Click here to edit and here for the organizers page. Please contact Frank to be added to the list of admins. Unless indicated otherwise, all talks start at 11:00 in LeConte 312. Tuesday, August 28 Speaker: Frank Thorne Title: Counting $S_3$-sextic fields Abstract: I will discuss my recent work with Takashi Taniguchi on counting $S_3$-sextic fields (i.e., sextic fields which are Galois over Q with Galois group $S_3$). I will start with a 30-minute silde talk I ll be giving at a conference soon, and then I ll go into some additional details at the board. (Please note: if someone else had designs on this date, I will happily yield the floor) Tuesday, September 4 Speaker: Joshua Harrington Title: The factorization of f(x)x^n+g(x) when deg < 3 Abstract: In this talk we investigate the factorization of the polynomials $f(x)x^n + g(x) \in \mathbb Z[x]$ in the special case where $f(x)$ is a monic quadratic polynomial with negative discriminant. We also mention similar results in the case that $f(x)$ is monic and linear. Page 1 of 9

2 Tuesday, September 11 Speaker: Ian Petrow (Stanford) Title: Continuous but nowhere differentiable functions which arise naturally in number theory Abstract: You probably recall functions like $f(x) = \sum_k \sin(k^2 x)/k^2$ from analysis 101 as the standard pathological examples of functions continuous on $\mathbb{r}$ but differentiable nowhere. This example might not worry you too much, however, since analysis 101 was probably also the last time you saw it. In this talk, I will explain how such functions turn up very naturally from character sums of the quadratic residue symbol. I will then discuss a second example I found of a continuous but nowhere differentiable function arising from a different arithmetic sum. It turns out that these scary nondifferentiable functions come from the theory of modular forms, particularly the theory of automorphic distributions and Eisenstein series. Saturday-Sunday, September Palmetto Number Theory Series, Wake Forest University, Winston-Salem, NC Invited Speakers: Noam Elkies, Sol Friedberg, Zev Klagsbrun, and Ian Petrow Funding is available for all participants. Tuesday, September 18 Speaker: Michael Filaseta Title: Page 2 of 9

3 Abstract: This will be a seminar about my new favorite number. Come and see why. This talk is on joint work with Sam Gross. Tuesday, September 25 Speaker: Richard Oh Title: Cryptanalysis of small valued secret exponents in RSA cryptosystems Abstract: In this talk, we will explore the vulnerability of RSA Cryptosystems due to poor implementation. In particular, when the secret exponent is small, a continued fractions algorithm will allow for factorization of the two large primes of the modulus, regardless of how well chosen they are. Furthermore, the algorithm used to expose this weakness runs in polynomial time. Joint work with Skip Garibaldi of Emory University. Tuesday, October 2 Speaker: Rodney Keaton (Clemson Univ.) Title: Level stripping of Siegel modular forms. Abstract: Let f be an elliptic eigenform of level Nl^a, where a is a positive integer, l is an odd prime, and N is a positive integer relatively prime to l. A result of Ribet gives the existence of an eigenform g of level N such that the eigenvalues of f away from the level remain congruent to the eigenvalues of g (mod l). This result was important in proving the equivalence of Serre's modularity conjecture and Serre's refined conjecture. As Herzig and Tilouine have recently made a Serre like conjecture for Siegel modular forms of genus 2, it is natural to ask if a level stripping result similar to Ribet's will hold in this setting. In this talk, after providing necessary background, we will present results in this direction. Page 3 of 9

4 Thursday, October 4, 3:30 in LC 412 Speaker: Steven Sam (UC-Berkeley) Title: Colloquium: Combinatorics and Geometry of E_7. Abstract: Exceptional objects can be thought of as an accident in classification schemes, but often have a rich structure all to themselves. In this talk, we'll explore some of the combinatorics and geometry related to the exceptional object E_7 (its root system, Weyl group, Lie algebra,...) which comes from the Cartan-Killing classification of simple Lie algebras. This object was studied by classical geometers long before this classification, and remains an object of interest today. We will discuss topics such as reflection arrangements, finite geometry, plane quartic curves, Kummer varieties, Vinberg's theta-representations, and toric geometry. The plan is to illustrate the beauty of this exceptional object in an accessible way. Tuesday, October 9 Speaker: Joshua Cooper (University of South Carolina) Title: On the Reciprocal of the Binary Generating Function for the Sum of Divisors Abstract: If $A$ is a set of natural numbers containing $0$, then there is a unique nonempty ``reciprocal'' set $A^{-1}$ of natural numbers such that every positive integer can be written in the form $a + a^\prime$, where $a \in A$ and $a^\prime \in A^{-1}$, in an even number of ways. It is straightforward to see that the generating functions of (the characteristic functions of) $A$ and $A^{-1}$ over $\mathbb{f}_2$ are reciprocals in $\mathbb{f}_2[[q]]$. Let $\Sigma$ denote the set containing $0$ and all positive integers such that $\sigma(n)$ is odd, where $\sigma(n)$ is the sum of all the positive divisors of $n$. Euler showed that $\sigma(n)$ satisfies an almost identical recurrence as that given by his Pentagonal Number Theorem, a corollary of which is that the set $P$ of natural numbers $n$ so that the partition function $p(n)$ is odd is the reciprocal of Page 4 of 9

5 the set of generalized pentagonal numbers (integers of the form $k(3k + 1)/2$, where $k$ is an integer). Therefore, motivated by the 1967 Parkin-Shanks Conjecture that the density of $P$ is $1/2$, we analyze the density $\rho$ of $\Sigma^{-1}$, conjecturing that $\rho = 1/32$ and proving that $0 \leq \rho \leq 1/16$. We also discuss a few surprising connections between $\Sigma$ and certain so-called ``Beatty sequences''. Joint work with Alex Riasanovsky of the University of Pennsylvania. Tuesday, October 16 Speaker: Daniel White Title: Strongly coloring pythagorean triples Abstract: It's easily shown that there exist $O(\log n)$-colorings of $\{1, \ldots, n\}$ such that no Pythagorean triple $\le n$ is monochromatic. One may consider the analogous problem of finding the number of colors required so that each Pythagorean triple $\le n$ is strongly colored. In particular, we investigate this number if one allows at most a vanishing proportion of Pythagorean triples $\le n$ to fail to have such a coloring. Covering systems are used as a main tool. Tuesday, October 23 in LC 412 Speaker: Lola Thompson (University of Georgia) Title: On the degrees of divisors of x^n-1 Abstract: We discuss what is known about the following questions concerning the degrees of divisors of x^n-1 in Z[x], as n ranges over the natural numbers: 1. How often does x^n-1 have AT LEAST ONE divisor of every degree between 1 and n? 2. How often does x^n-1 have AT MOST ONE divisor of every degree between 1 and n? Page 5 of 9

6 3. How often does x^n-1 have EXACTLY ONE divisor of every degree between 1 and n? 4. For a given m, how often does x^n-1 have a divisor of degree m? We will also discuss what changes when Z is replaced by the finite field F_p. A portion of this talk is based on joint work with Paul Pollack. Tuesday, October 30 Speaker: John Willis Title: An overview of BSD, I. Abstract: We will discuss the objects necessary to understand the proof of the Gross-Zagier/Kolyvalgin theorem, which gives us the only known cases of the Birch and Swinnerton-Dyer conjecture. In particular, we will discuss the Shafarevich-Tate group, modular parameterizations of elliptic curves, complex multiplication, and Heegner points with an eye towards the Gross-Zagier formula and Heegner systems, which are indispensable tools in the proof of this result. Tuesday, November 6 (USC Holiday) Tuesday, November 13 Speaker: Ari Shnidman (Michigan) Title: Counting cubic number fields and related problems Abstract: The Davenport-Heilbronn theorem gives asymptotics for the number of cubic fields having bounded discriminant. I will explain an elementary way to count Page 6 of 9

7 $C_3$-cubic fields, i.e. those which are Galois over the rationals. More generally, we give asymptotics for the number of cubic fields having a fixed quadratic resolvent field. Our approach also lets us count orders in cubic fields. If there is time, I will discuss an application of these results to bounding the average rank of elliptic curves in families of cubic twists. This is joint work with Manjul Bhargava. Monday, November 19 in LC 412 (Department Colloquium) Speaker: Ravi Vakil (Stanford) Title: Cutting and pasting in algebraic geometry Abstract: Given some class of "geometric spaces", we can make a ring as follows. additive structure: When U is an open subset of such a space X, [X] = [U] + [X - U] multiplicative structure: [X x Y] = [X][Y]. In the algebraic setting, this ring contains surprising structure, connecting geometry to arithmetic and topology. I will discuss some remarkable statements about this ring (both known and conjectural), and present new statements (again, both known and conjectural). A motivating example will be polynomials in one variable. (This talk is intended for a broad audience.) This is joint work with Melanie Matchett Wood. Tuesday, November 20 Speaker: Cancelled. Be sure to see Ravi s talk instead. Tuesday, November 27 Speaker: John Willis. Title: An overview of BSD, II. Page 7 of 9

8 Abstract: John will finish his proof of the Birch and Swinnerton-Dyer Conjecture. Friday, November 30 Speaker: Carl Pomerance (Dartmouth) Title: TBA (Special Colloquium Lecture) Abstract: TBA. Saturday-Sunday, December 1-2 Palmetto Number Theory Series, USC Invited Speakers: Benedict Gross, Winfried Kohnen, Carl Pomerance, Matthew Young, Wei Ho, Alyson Deines Funding is available for all visitors. Tuesday, December 4 Speaker: Jayce Getz (Duke) Title: An approach to nonsolvable base change for GL(2) Abstract: Motivated by Langlands' beyond endoscopy idea, the speaker will present a conjectural trace identity that is essentially equivalent to base change and descent of automorphic representations of GL(2) along a non-solvable extension of fields. Page 8 of 9

9 Page 9 of 9