This seminar is held on Tuesdays at 10am in 104 Deady Hall unless otherwise stated.

Spring Quarter, 2018

• April 10, Asif Zaman (Stanford)
  Primes represented by positive definite binary quadratic forms
  Abstract: The distribution of primes represented by positive definite integral binary quadratic forms is a classical topic within number theory and has been intensely studied over many centuries. We will briefly describe some of its history and discuss results giving new upper bounds for the number of primes up to $x$ represented by a given form. These estimates will come in several different flavours: unconditional, conditional on the Generalized Riemann Hypothesis, and on average over discriminants. The key feature will be that non-trivial bounds are obtained when the size of $x$ is a small power of the discriminant and, in many cases, this size will be essentially optimal.
• April 10, Florian Sprung (Arizona State University)
  Special Time/Location: 11am in Deady 301
  The Iwasawa Main Conjecture for Elliptic Curves at Supersingular Primes
  Abstract: Iwasawa theory is a connection between algebra and analysis. In the context of elliptic curves, the main conjecture of Iwasawa theory states that an analytically constructed p-adic L-function matches its algebraic counterpart. This conjecture is known when p is an ordinary prime. In this talk, we will explain our results in the case where the prime is a general supersingular prime.
• May 17, Kevin McGown (California State University, Chico)
  Special Time/Location: 3 pm in Anstett 192
  Grosswald’s conjecture on the least primitive root
• May 19, Rachel Pries (Colorado State University)
  Speaking at Oregon Number Theory Days
• May 22, Aaron Pollack (Duke University)
  Special Time/Location : 11am in Deady 301
  The Fourier expansion of modular forms on exceptional groups
  Abstract: The classical group Sp(2n) has a nice notion of classical modular forms, namely, the holomorphic Siegel modular forms. However, exceptional groups such as G_2, F_4, and E_8 do not obviously have any classical “modular forms,” because their associated symmetric spaces don’t have complex structure. Nevertheless, Gross-Wallach and Gan-Gross-Savin have defined a notion of modular forms on these groups, and (using key input of Wallach) proved abstractly that these modular forms admit a nice Fourier expansion. I will explain what these modular forms are, and what their Fourier expansions look like explicitly.

Winter Quarter, 2018

• January 27, Ken Ono (Emory University)
  Speaking at Oregon Number Theory Days
• February 6, Marty Weissman (UCSC)
  An introduction to metaplectic groups
  Abstract: In his 1964 Acta paper, André Weil introduced metaplectic groups. For Weil, these were groups generated by certain unitary operators on a space of L2 functions. His paper brought together harmonic analysis and number theory, yielding new results on quadratic forms and a proof of quadratic reciprocity. Within about 10 years, Shimura had carried out a deep study of modular forms of half-integer weight and Gelbart and Piatetski-Shapiro linked half-integer weight modular forms to automorphic forms on Weil’s metaplectic groups.

  The metaplectic groups are central extensions of symplectic groups by a group of order 2. Soon after Weil’s analytic construction, Steinberg and Matsumoto studied central extensions of Chevalley groups (like SL_n) over arbitrary fields, finding a link to algebraic K-theory. This provided an algebraic approach to metaplectic groups, and a broader class of groups to study with applications to automorphic forms and number theory.

  In this talk, I will give a historical introduction to the metaplectic group and its generalizations, focusing on algebraic aspects and finishing with Brylinski and Deligne’s category of “central extensions of reductive groups by K2”. No familiarity with algebraic groups, K-theory, or automorphic forms is required. The style will be comparable to a “What is…” paper in the Notices.

• February 13, Marty Weissman (UCSC)
  Extending the Langlands program to covering groups
  Abstract: Among the very first modular forms, studied by Jacobi, were modular forms of half-integer weight. In modern terms, these can be viewed as automorphic forms on the metaplectic group. Since the metaplectic group is not an algebraic group, the conjectures of Langlands do not apply — Langlands did not conjecture a relationship between automorphic representations of metaplectic groups and Galois representations, for example.

  I will describe recent efforts to close this basic gap in the Langlands program, by constructing an “L-group” for a broad class of covering groups (including the metaplectic group). This L-group allows one to reformulate Langlands conjectures for covering groups. I will discuss the classification of covering groups (after Brylinski-Deligne), the construction of this L-group, and evidence for an extension of the Langlands program.

• February 27, Sarah Frei (UO)
  Moduli spaces of sheaves on a K3 surface and Galois representations
  Abstract: For a fixed K3 surface, we can study various moduli spaces of sheaves having a given topological type. Under appropriate conditions on the topological data, the moduli spaces are smooth projective varieties. I will discuss a new result about the geometry and arithmetic of these moduli spaces when the K3 surface is defined over an arbitrary field. For any two such moduli spaces of the same dimension, their étale cohomology groups with -coefficients are isomorphic as Galois representations. In particular, when the K3 surface is defined over a finite field, this implies that the moduli spaces have equal zeta functions.
• March 13, Robert Hines (University of Colorado, Boulder)
  Badly approximable numbers over imaginary quadratic fields
  Abstract: We will recall two characterizations of badly approximable real numbers (via continued fractions and geodesic trajectories on the modular surface) and discuss real quadratic irrationals from both perspectives. We will then generalize these characterizations to imaginary quadratic fields (using continued fractions when applicable and geodesic trajectories in the Bianchi orbifolds more generally). Finally we will use these characterizations to produce examples of badly approximable numbers (those lying on anisotropic rational circles). These examples include algebraic numbers of every even degree over the rationals.

Fall Quarter, 2017

• October 21, Kirsten Eisentraeger (Penn State University)
  Oregon Number Theory Days
• November 7, Christopher Pinner (Kansas State University)
  The Lind-Lehmer Constant for Finite Abelian Groups
  Abstract: In 2005 Doug Lind generalized the concept of Mahler measure to an arbitrary compact abelian group. For a given group one can ask for the minimal non-trivial measure; the counterpart of the classical Lehmer Problem for the usual Mahler measure. For a finite abelian group this corresponds to the smallest non-trivial integral group determinant. After a quick survey of existing results I will present some new congruences satisfied by the Lind Mahler measure for p-groups. These enable us to determine the minimal measure when the p-group has one particularly large component and to compute the minimal measures for many new families of small p-groups.

  This is joint work with Mike Mossinghoff of Davison College. If there is time I will also mention some 3-group results from a summer undergraduate research project with Stian Clem which may hint at what is going on in general.

• November 9, Bianca Viray (University of Washington)
  Special Time/Location: 10am in Deady 210
  Points of unusually low degree on the modular curves X_1(n)
  Abstract: The Mordell-Weil theorem says that for any elliptic curve E over a number field K, the set E(K) is a finitely generated abelian group. In particular, the torsion subgroup of E(K) is finite. Amazingly, the size of the torsion subgroup can be bounded by a constant that depends only on [K:Q], i.e., that is independent of K and E; this is Merel’s celebrated Uniform Boundedness Theorem. This theorem can be rephrased as saying that for any positive integer d, the infinite tower of modular curves {X_1(n)}_{n} has only finitely many closed points of degree at most d. Work of Frey and Abramovich from around the same time combine to give an independent proof of a weaker result, that for any positive integer d, there are only finitely many positive integers n such that X_1(n) has infinitely many degree at most d points. Moreover, the work of Frey essentially characterizes these integers n in terms of the gonality of X_1(n) and the Q-rational points on Jac(X_1(n)). In this talk, we study complementary part of Merel’s theorem, that is, the modular curves X_1(n) which have finitely many (but nonempty) points of degree at most d.
• November 14, Yifeng Liu (Northwestern University)
  Special cycles, level raising, and Selmer groups
  Abstract: In this talk, we will introduce recent progress on the Bloch-Kato conjecture about L-functions and Selmer groups. We will explain how the arithmetic geometry of Shimura varieties and special cycles is used in the problem through ​congruence of automorphic forms, known as level raising.

Previous years: 2017 2016