Scroll down for abstracts for the most recent seminars
Thurs. Jan. 25----Dongho Byeon (Seoul National University)
Title: Rank-one quadratic twists of an infinite familly of elliptic curves
Abstract: A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E has (analytic) rank 1. This assertion has been confirmed by Vatsal and Byeon for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.
Thurs. Feb. 1----Jeff Achter (Colorado State)
Title: Monodromy of hyperelliptic and trielliptic curves
Abstract: I'll explain a calculation of the integral monodromy of every component of the moduli spaces of hyperelliptic and trielliptic curves (joint work with R. Pries). From this I'll deduce information about class groups of function fields, including a result of Cohen-Lenstra type for quadratic function fields.
Thurs. Feb. 8----David Penniston (Furman)
Title: Divisibility and distribution of some arithmetic functions
Abstract: In this talk we investigate the congruence properties of several arithmetic functions, including some partition functions and the Fourier coefficients of certain harmonic weak Maass forms.
Tues. Feb. 13*----Frank Thorne (UW-Madison)
Title: Mordell-Weil ranks in families of elliptic curves
Abstract: In this talk, we will survey some conjectures and results concerning the distribution of ranks of elliptic curves. We will start by giving a precise description of what our question means, and discuss some related basic principles. We then give a brief overview of the "Katz-Sarnak philosphy" used to make conjectures about these distributions, and conclude with a brief discussion of several results about them, due to Heath-Brown, Gouvea and Mazur, and Ono and Skinner.
Thurs. Feb. 15----William Hoyt
Title: Arithmetic for twisted Legendre equations associated with six lines in the plane
Abstract: Correspondences will be described between modular families of Abelian surfaces and families of twisted Legendre equations which determine elliptic fiberings of corresponding Kummer surfaces. These families correspond to configurations of 6 lines in the plane with at least one triple vertex and (in almost all cases) with suitable extra contact with suitable rational curves. In particular, twisted Legendre equations of the form yy=t(t-1)(t-c)x(x-1)(x-t) with c not equal to 0,1 correspond to products ExE of elliptic curves with equations of the form ww=4zzz-3(c-3/4)z-(c+9/8) ; the first equation has a C(t)-rational solution (x(t),y(t)) with y not equal to 0 if and only if E has complex multiplication; and the form of x(t) determines a Q-rational condition on c which seems (?) to be equivalent to the class equation for a corresponding special value of a Weber function.
Thurs. Feb. 22----Matt Darnall (UW-Madison)
Title: Bounds on the Star discrepancy in the Unit Square
Abstract: (t,m,s) nets are sets of points in the unit cube that are "well distributed" in a precise sense, i.e they achieve the best known star discrepancy. In this talk, I will show that the discrepancy at any point of a random (0,m,2) net in base 2 behaves no worse than a simple symmetric random walk with O(m) steps.
Thurs. March 1----Maosheng Xiong (UIUC)
Title: Selmer groups and Tate-Shafarevich groups for the congruent number problem
Abstract: We study the distribution of the sizes of the Selmer groups arising from the three 2-isogenies and their dual 2-isogenies for the elliptic curve $E_n:y^2=x^3-n^2x$. We show that three of them are almost always trivial, while the 2-rank of the other three follows a Gaussian distribution. It implies three almost always trivial Tate-Shafarevich groups and three large Tate-Shararevich groups. When combined with a result obtained by Heath-Brown, we show that the mean value of the 2-rank of the large Tate-Shafarevich groups for square-free positive odd integers $n \le X$ is $\frac{1}{2} \log\log X+O(1)$, as $X \to \infty$.
Thurs. March 8----Frank Calegari (Northwestern)
Title: Deformations of Galois representations over arbitrary number fields
Abstract: The theory of two dimensional p-adic Galois representations of the absolute Galois group of Q (and their relation to modular forms) is well studied. By now, a standard approach to studying such objects is to place both in "p-adic families". Many of these methods also work when Q is replaced by a totally real field K. We talk about the general case, and explain why it should look substantially different. We discuss why on expects there to be "very few" modular forms over K, and where to find the "missing" automorphic forms corresponding to the Galois representations. This is joint work with Barry Mazur.
Thurs. March 15----Alina Cojocaru (UIUC)
Title: Twin primes for elliptic curves
Abstract: Let E be an elliptic curve over Q and let E_p be its reduction modulo a prime p. I will discuss the problem of finding an asymptotic formula for the number of primes p < x for which E_p has prime order. In particular,
I will discuss an average version of this problem (joint work with Antal Balog and Chantal David).
Thurs. March 22----Paul Garrett (Minnesota)
Title: Integral moments of automorphic L-functions, and applications
Abstract: We produce integral moment identities for GL(m)xGL(n) automorphic L-functions on the critical line, from natural representation theoretic considerations, giving these integral moments instrinsic structural meaning. Spectral expansions of the associated Poincare' series give meromorphic continuations of the integral moments. We find that the natural form of higher moments (beyond the second, for GL(2), or beyond the fourth, for GL(1), for example) differs from the traditional formulation. To shed light on this, we incidentally show that the traditional formulation of higher moments has the defect that the associated generating function has a natural boundary (Estermann phenomenon). A sample application of the identities for second moments, with concommitant asymptotics, is to t-aspect subconvexity. For GL(2)xGL(1), convexity-breaking follows by standard methods from asymptotics with non-trivial error terms. The GL(2) case extends classical work of Good, Ivic, Jutila, and Motohashi, for groundfield Q, to arbitrary number fields and arbitrary archimedean types.
Thurs. March 29----Rafe Jones (UW-Madison)
Title: Iterated endomorphisms of abelian algebraic groups
Abstract: Let V be a variety defined over Q, A a point in V(Q), and phi : V --> V a nice morphism (e.g. finite, defined over Q, surjective on V(Qbar)). The preimage tree of A consists of the points in the sets { phi^{-n}(A) }, n > 0, which form a tree if one assigns edges via the action of phi. The absolute Galois group of Q acts on T as tree automorphisms, producing an "arboreal Galois representation." In general the image of this representation should be a large subgroup of Aut(T), though this remains highly conjectural. In this talk we consider a case where the image is a small subgroup of Aut(T), and thus more manageable, namely the case where V is an abelian algebraic group and phi is multiplication by a prime l. The image in this case encodes information about the density of p such that l divides the order of the mod p reduction of A. When V is relatively simple, ie of dimension 1 or an untwisted torus, then this density can actually be calculated. A sample result is that if V is a sufficiently general elliptic curve and A is nontorsion and not in 2E(Q) then the reduction mod p of A has even order for 11/21 of all p. I will discuss similar results in the case of tori and CM elliptic curves, and give some applications. The first part of the talk will contain some examples of arboreal representations in the case V = P^1 and phi a quadratic polynomial, where several basic questions are still unknown.
Thurs. April 12----Jan Bruinier (University of Cologne)
Title: Twisted Borcherds products on modular curves
Abstract: We discuss some generalizations of the regularized theta liftings of Borcherds, Harvey, and Moore.
Thurs. April 26----George Andrews (Penn State)
Title: Partitions, Durfee Symbols and the Atkin-Garvan Moments of Ranks
Abstract: In 1944, Freeman Dyson defined the rank of a partition with the object of providing a combinatorial interpretation of the Ramanujan conguences for the partition function, p(n). Dyson's discoveries and conjectures have led to an extensive field of research with exciting discoveries by Atkin, Garvan, Swinnerton-Dyer, Ono, Bringmann and Mahlburg and others. In this talk, we consider moments that Atkin and Garvan associated with the ranks defined by Dyson. We shall reveal the objects they enumerate and shall discuss implications and possibilities. We conclude with a brief discussion of spt(n), the number of smallest parts in the partitions of n, and shall relate this function to the Atkin-Garvan moments as well. As somewhat of a surprise, we can prove that 13 | spt(13n+6).
Thurs. May 3---- Amanda Folsom (UW-Madison)
Title: Duality involving the mock theta function $f(q)$ and analytic properties of Kloosterman-Selberg zeta functions
Abstract: We show that the coefficients of Ramanujan's mock theta function $f(q)$ are the first nontrivial coefficients of a canonical sequence of modular forms. This fact follows from a duality which equates coefficients of the holomorphic projections of certain weight 1/2 Maass forms with coefficients of certain weight 3/2 modular forms. This work depends on the theory of Poincar\'e series, and a modification of an argument of Goldfeld and Sarnak on Kloosterman-Selberg zeta functions.
Thurs. May 11---- David Solomon (King's College)
Title: L-functions at s=1 and Reciprocity Laws for Rubin-Stark Elements
Abstract: Let k be a totally real number field of degree r, L an abelian, CM extension of k with group G and S an appropriate finite set of places of L. Stark's conjecture predicts that the for all even characters \chi of G, the rth derivatives of the S-truncated L-functions L_S(s,\chi) at s=0 are determined by an equivariant rxr regulator of certain S-units of L^+. These generalise cyclotomic units (the case k=Q) but are not known to exist in many other nontrivial cases. For odd characters however, L_S(0,\chi) defines an ideal in the `minus part' ZG^- of ZG that generalises the Stickelberger ideal (the case k=Q) and is conjectured to annihilate the class group of L. In the same set-up I'll show how to use L_S(1,\chi) for odd \chi to define a map \mathfrak{s} from exterior powers of p-semilocal units into Q_pG^- (p an odd prime). The image appears to be a new p-adic analogue of the Stickelberger ideal and I conjecture that it lies in Z_pG^-. If L contains p^nth roots of 1, I conjecture that further that the map \mathfrak{s} (coming from odd characters at s=1) gives a sort of explicit reciprocity law for the above mentioned S-units of L^+ (coming from even characters at s=0).