Thurs. Sept. 8----Nigel Boston (UW-Madison)
Title: Galois groups of tamely ramified p-extensions.
Abstract: Explicit presentations of the Galois group of the maximal p-extension of Q unramified outside a finite set S of primes not containing p, have eluded researchers for many years. This talk will report on work by the speaker, by Labute, and others describing these groups and their properties, with striking consequences and connections to algebraic topology, root-discriminant problems, deficiency zero problems, and the Fontaine-Mazur conjecture.
Thurs. Sept. 15---Andrew Shallue (UW-Madison)
Title: Deterministically finding points on elliptic curves over finite fields.
Abstract: In this talk we'll consider the computational problem of finding points on elliptic curves over finite fields. This has cryptographic applications; for example, the ElGamal encryption system requires a message to be encoded as a point on an elliptic curve. An easy probabilistic polynomial-time algorithm exists to solve the problem. Given an elliptic curve E[F_p]: y^2 = x^3 + ax + b with p prime and > 3, pick a random x. If the Legendre symbol (x^3+ax+b / p ) is 1, take the square root to get the point (x,y) in E. Otherwise try another random x. A similar probabilistic algorithm exists for elliptic curves over finite fields of characteristic 2. In this talk we'll give a deterministic polynomial-time algorithm for finding points on elliptic curves with j-invariant 0 over finite fields of characteristic 2.
Thurs. Sept. 22---Ben Brubaker (Stanford)
Title: Recent Progress on Dirichlet Series Associated to Weyl Groups
Abstract: In a series of papers with Dan Bump and others, we prove the meromorphic continuation and functional equations for a family of Dirichlet series in several variables. The groups of functional equations are Weyl groups. Moreover, these Dirichlet series are conjectured to arise from the Whittaker coefficients of certain Eisenstein series attached to simple Lie groups with the associated Weyl group. In this talk, we'll discuss what sorts of Dirichlet series arise, giving concrete examples, and then survey the arithmetic applications (both past and future) which can/may be obtained from them. (To the indoctrinated: Efforts will be made to discuss different, but related, topics from those arising in the Bretton Woods workshop talks.)
Thurs. Sept. 29 -- Ken Ono (UW-Madison)
Title: Modular Forms, Infinite Products, and Singular Moduli
Abstract: Modular forms play many roles in mathematics. In number theory, modular forms often arise as generating functions for interesting quantities such as representation numbers of integers by quadratic forms, partition functions, values of L-functions, and also degrees of characters of sporadic simple groups like the Monster. In his 1994 ICM lecture, Borcherds found a striking new phenomenon. He proved that certain modular forms of half-integral weight serve as generating functions for the infinite product exponents of other modular forms, thereby greatly generalizing some of the prettiest q-series dating back to works of Euler and Jacobi on classical theta functions. His work pertained to an exceptionally rich family of modular forms, those with a `Heegner divisor'. Zagier later found a beautiful number theoretic explanation of the Borcherds phenomenon, one involvingsingular moduli, complex multiplication, and elliptic curves. In this lecture, we provide a general framework which includes Zagier's reformulation of Borcherds' theory as a special case. We show that all of these results follow from beautiful properties of a delightfully rich sequence of modular forms, the weak Maass-Poincare series of half-integral weight.
Thurs. Oct. 13 -- Jordan Ellenberg (UW-Madison)
Title: Speculations about asymptotic formulas for number of rational points on varieties over number fields and function fields
Abstract: Let K be a number field. The Batyrev-Manin heuristic proposes an asymptotic formula for the number of rational points in X(K) of height at most B. On the other hand, the Malle conjecture proposes a very similar asymptotic formula for the number of degree-d extensions of K of discriminant at most B. We'll discuss what is known about these conjectures at present and why one might consider them analogous; we will then discuss some ideas about the geometry that might underly the conjectures (or which might, on the other hand, lead us to doubt them!) when K is taken to be a function field over a finite field; in particular, we will try to suggest a connection between these questions and purely topological theorems about stable cohomology of certain families of moduli spaces.
Connections will be drawn with material from Math 847, Jeremy Rouse's talks on the Weil conjectures, and the stacks seminar, so the audience for these talks is encouraged to come to this talk! (but attendance at the other talks will not be assumed...)
Tuesday Oct. 18 -- Hui Xue (Michigan)
Title: Central values of L-functions over CM fields
Abstract: We will show an explicit formula for the central value of certain Rankin $L$-function $L(\pi\otimes{\chi})$, where $\pi$ is an automorphic representation associated to a Hilbert newform $f$ and ${\chi}$ is a unitary anticyclotomic Hecke character of a CM field. As corollaries some arithmetic properties of central values can be obtained. The proof uses explicit computations which involve theta correspondences.
Thurs. Oct. 20 -- Rafe Jones (UW-Madison)
Title: The density of prime divisors of quadratic recurrences
Abstract: Given a recurrence sequence a_n, the set of primes dividing a_n is a natural object of study; consider for instance the Fermat numbers 2^{2^n}+1 or the Mersenne numbers 2^n-1. While this set of primes is difficult to describe precisely, there has been considerable work on determining its density. This density has been determined for large classes of linear recurrences, but little attention has been paid to higher degree recurrences. In this talk I'll present some results on quadratic recurrences of the form a_n = f(a_{n-1}), where f is a monic quadratic polynomial with integer coefficients. The density of prime divisors in this case depends on properties of the Galois tower generated by iterates of f. This tower in turn has much to do with arithmetic properties of the forward orbit of the critical point of f. This makes for a striking analogy with recent results in real and complex dynamics, where properties of the forward orbit of the critical point have been shown to essentially determine the dynamics of quadratic polynomials.
Thursday, Oct. 27 -- Dave Roberts (U. Minnesota-Morris)
Title: 2-adic ramification in some 2-extensions of Q
Abstract: Let G_p be the pro-2-group classifying 2-extensions of Q with absolute discriminant of the form 2^a p^b. If p is congruent to 3 or 5 modulo 8 then it is known that G_p coincides with its 2-adic decomposition group, so that G_p itself is filtered by ramification subgroups measuring wildness of 2-adic ramification. We restrict to this case and study the filtration on G_p. It is known that the isomorphism class of G_p depends only on p modulo 8. Our computations suggest that even the filtration may depend only on p modulo 8.
Thursday, Nov. 3 -- Tal Sutton (UW-Madison)
Title: Theta Liftings and Central Values of Automorphic L-Functions
Abstract: Let $\sigma$ be a an irreducible, cuspidal, automorphic representation on the metaplectic group and $\psi$ an unramified additive character of the adeles. When the root number of a certain representation of $PGL_2$ is 1, Waldspurger proved that the nonvanishing of an $L$-function associated to $\sigma$ at it's critical value is equivalent to the nontriviality of the global theta lift of $\sigma$ with respect to $\psi$. Using a certain linear operator of $\sigma$ we provide a new proof of this theorem of Waldspurger's and discuss how, in certain cases, this linear operator can be used to explicitely compute the $L$-function at its critical value.
Tuesday, Nov. 8 -- Adam Logan (U. Liverpool/CRM)
Title: Descent by Richelot isogeny on the Jacobians of plane quartics
Abstract: In this talk I will discuss practical and theoretical features of complete 2-descent and 2-descent by isogeny on the Jacobians of curves of genus 3. In particular, I will give (what I believe to be) the first example of a smooth plane quartic with irreducible Jacobian and no nontrivial automorphisms whose Jacobian has nontrivial Tate-Shafarevich group. The tools used include Schaefer's method for computing Selmer groups of Jacobians, Bruin's ideas on plane sections of Kummer surfaces, and the Donagi-Livn\'e construction of curves of genus 3 with isogenous Jacobians.
Thursday, Nov. 10 -- Jeremy Rouse (UW-Madison)
Title: Traces of Singular Moduli on Hilbert Modular Surfaces
Abstract: Suppose that p = 1 mod 4 is prime
and let K =
Q(p^(1/2)). Hirzebruch and Zagier proved that generating
functions for the intersection numbers of Hirzebruch-Zagier divisors on
the Hilbert modular
surface (H x H)/SL(2,O_K) are weight 2 modular forms. Using work of
Bruinier and Funke, we show that generating functions for traces of
singular moduli over these intersections are weakly holomorphic weight
2 modular forms. For the singular moduli of j(z) - 744 we compute these
generating functions explicitly, and factorize their "norms" as
products of Hilbert class polynomials.
Thursday,
Nov. 17 -- Doug Ulmer (Arizona)
Title: Ranks of Jacobians over function fields
Abstract: The Jacobian J of an algebraic curve C measures the obstruction to writing down a function on C with zeroes and poles at prescribed points. When the ground field is the complex numbers, J has a very explicit description in terms of line integrals on the curve. When the ground field k is finitely generated (e.g., a number field or function field over a finite field), then the set of rational points J(k) is a finitely generated abelian group. In this talk we will write down explicit curves of genus g over the rational function field k=Fp(t) (every p, every g>0) for which the rank of J(k) is arbitrarily large and for which the BSD conjecture holds. Quite a bit of background possibly useful for grad students will be given.
Thursday, Dec. 1 -- Rachel Pries (Colorado State)
Title: p-torsion of curves in characteristic p
Abstract: Over the complex numbers, an
elliptic curve has $p^2$ points of order $p$ and a curve of genus $g$
has $p^{2g}$ points of order $p$ on its Jacobian. The
situation is different in characteristic $p$ where an elliptic curve
has at most $p$ points of order $p$ and a curve of genus $g$ has
$p^{f}$ points of order $p$ on its Jacobian where $0 \leq f \leq g$.
The invariant $f$ is the {\it p-rank} of the curve.
I will talk about joint work with Darren Glass on the p-ranks
of hyperelliptic curves.
Invariants of abelian varieties in characteristic $p$ such as the
$p$-rank are well-understood and have been used to define
stratifications of the moduli space $A_g$ of principally polarized
abelian varieties of dimension $g$. There is a deep interest
in understanding whether the Torelli locus intersects these strata in
$A_g$. More generally, one can ask for the dimension of the
intersection of these strata with the image of the moduli spaces $M_g$
or $H_g$ under the Torelli map. I
will also talk about joint work with Darren Glass in which we show that
the Torelli locus intersects several of these strata by producing
families of curves so that the $p$-torsion of the Jacobian of each
fibre contains certain group schemes.
Thursday, Dec. 1 -- Kathrin Bringmann (UW-Madison)
Title: Freeman Dyson's "Challenge for the
Future": The mock theta functions.
Abstract: In his last letter to Hardy, Ramanujan defined 17 peculiar
functions which are now referred to as his mock theta functions.
Although these mysterious functions have been investigated by many
mathematicians over the years, many of their most basic properties
remain unknown. This inspired Freeman Dyson to proclaim:
"The mock theta-functions give us tantalizing hints of a grand
synthesis still to be discovered. Somehow it should be possible to
build them into a coherent group-theoretical structure, analogous to
the structure of modular forms which Hecke built around the old
theta-functions of Jacobi. This remains a challenge for the future."
Freeman Dyson
1987, Ramanujan Centenary Conference
Here we announce a solution to Dyson's "challenge for the future" by
providing the "coherent group-theoretical structure" that Dyson desired
in his plenary address at the 1987 Ramanujan Centenary Conference.
In joint work with Ken Ono, we show that Ramanujan's mock theta
functions, as well a natural generalized infinite class of mock theta
functions may be completed to obtain Maass forms, a special class of
modular forms. We then use these results to prove theorems
about Dyson's partition ranks. In particular, we shall prove
the 1966 Andrews-Dragonette Conjecture, whose history dates to
Ramanujan's last letter to Hardy, and we shall also prove that Dyson's
ranks `explain' Ramanujan's partition congruences in an unexpected way.