NTS ABSTRACTFall2018
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Sept 6
Simon Marshall |
What I did in my holidays |
Abstract: I will discuss progress I made on the subconvexity problem for automorphic L-functions while at the IAS this past year. |
Sept 13
Nigel Boston |
2-class towers of cyclic cubic fields |
Abstract: The Galois group of the p-class tower of a number field K is a somewhat mysterious group. With Bush and Hajir, I introduced heuristics for how often this group is isomorphic to a given finite p-group G (p odd) as K runs through all imaginary (resp. real) quadratic fields. The next case of interest is to let K run through all cyclic cubic fields. The case p=2 already introduces new phenomena as regards the distribution of p-class groups (or G^{ab}), because of the presence of pth roots of 1 in K. Our investigations indicate further new phenomena when investigating the distribution of p-class tower groups G in this case. Joint work with Michael Bush. |
Sept 20
Naser T. Sardari |
Bounds on the multiplicity of the Hecke eigenvalues |
Abstract: Fix an integer N and a prime p\nmid N where p> 3. Given any p-adic valuation v_p on \bar{\mathbb{Q}} (normalized with v_p(p)=1) and an algebraic integer \lambda \in \bar{\mathbb{Q}}; e.g., \lambda=0, we show that the number of newforms f of level N and even weight k such that T_p(f)=\lambda f is bounded independently of k and only depends on v_p(\lambda) and N. |
Sept 27
Florian Ian Sprung |
How does the Mordell-Weil rank of an elliptic curve grow in towers of number fields, if you start with a quadratic imaginary field? |
Abstract: The Mordell-Weil theorem says that for a number field $K$, the $K$-rational solutions of an elliptic curve $E: y^2=x^3+ax+b$ form a finitely generated abelian group. One natural question is how the rank of these groups behave as $K$ varies. In infinite towers of number fields, one may guess that the rank should keep growing ('more numbers should mean new solutions'). However, this guess is not correct: When starting with $\Q$ and climbing along the tower of number fields that make up the $Z_p$-extension of $Q$, the rank stops growing, as envisioned by Mazur in the 1970's.
What happens if we start with an imaginary quadratic field instead of $\Q$, and look at the growth of the rank in a $Z_p$-extension? Our new found 'intuition' now tells us we should expect the rank to stay bounded, but this is not always the case, as shown in Bertolini's thesis. So how badly does the rank grow in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei. |
Oct 4
Renee Bell |
Local-to-Global Extensions for Wildly Ramified Covers of Curves |
Abstract: Given a Galois cover of curves $X \to Y$ with Galois group $G$ which is totally ramified at a point $x$ and unramified elsewhere, restriction to the punctured formal neighborhood of $x$ induces a Galois extension of Laurent series rings $k((u))/k((t))$. If we fix a base curve $Y$, we can ask when a Galois extension of Laurent series rings comes from a global cover of $Y$ in this way. Harbater proved that over a separably closed field, every Laurent series extension comes from a global cover for any base curve if $G$ is a $p$-group, and he gave a condition for the uniqueness of such an extension. Using a generalization of Artin--Schreier theory to non-abelian $p$-groups, we characterize the curves $Y$ for which this extension property holds and for which it is unique up to isomorphism, but over a more general ground field. |
Oct 11
Chen Wan |
A Local Trace Formula for the Generalized Shalika model |
Abstract: I will discuss the local multiplicity problem for the generalized Shalika model. By proving a local trace formula for the model, we are able to prove a multiplicity formula for discrete series, which implies that the multiplicity of the generalized Shalika model is a constant over every discrete local Vogan L-packet. I will also discuss the relation between the multiplicity and the local exterior square L-function. This is a joint work with Rapheal Beuzart-Plessis. |
Oct 18
Mark Shusterman |
The fundamental group of a smooth projective curve over a finite field is finitely presented |
Abstract: Solutions of (sets of) polynomial equations are (for quite some time) studied using the geometry of the associated varieties. The geometric approach was very successful, for instance, in the case of curves over finite fields.
Associated to a curve is its (etale) fundamental group. This is a mysterious profinite group that ‘remembers’ the count of solutions to the equations giving rise to the curve, and sometimes also the curve itself (up to isomorphism). Grothendieck, using fundamental groups of complex curves, shed light on these mysterious profinite groups, showing (in particular) that they are finitely generated. We will show that these groups are furthermore finitely presented, hoping to find a finitary description for them (one day). |
Oct 25
Mark Shusterman | |
The fundamental group of a smooth projective curve over a finite field is finitely presented | |
Let K be the function field of a curve C over a finite field Fq, and
let A be an abelian variety over K. Analogy with the classical BrauerSiegel theorem led Hindry to consider the ratio BS(A) := log (|X(A)|Reg(A)) log H(A) where |X(A)| is the order of the Tate-Shafarevich group of A, Reg(A) is the Néron-Tate regulator of A, and H(A) is the exponential differential height of A. If An, n ≥ 1 is a sequence of abelian varieties of fixed dimension over K with H(An) → ∞, the classical analogy would suggest that limn→∞ BS(An) = 1. Hindry, Pacheco, and Griffon have used analytic techniques to give several example of families {An} for which the limit above exists and is equal to 1. (They also gave evidence for the conjecture that a limit of zero is possible.) More specifically, they use the formula of Birch and Swinnerton-Dyer (in cases where it is known to hold) and an estimation of the leading Taylor coefficient of the L-function of An over K to estimate the size of |X(A)|Reg(A). In this talk, I will explain an algebraic technique for estimating |
Reg(A) directly which recovers the limit above in all cases considered
by Hindry, Pacheco, and Griffon. One technical novelty in our approach is that we use properties of the function m 7→ |X(A/Fqm(C))| to estimate the size of |X(A)|Reg(A) over K = Fq(C), i.e., over the original ground field. |
Nov 1
Jinbo Ren |
Mathematical logic and its applications in number theory |
Abstract: A large family of classical problems in number theory such as:
a) Finding rational solutions of the so-called trigonometric Diophantine equation $F(\cos 2\pi x_i, \sin 2\pi x_i)=0$, where $F$ is an irreducible multivariate polynomial with rational coefficients; b) Determining all $\lambda \in \mathbb{C}$ such that $(2,\sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are both torsion points of the elliptic curve $y^2=x(x-1)(x-\lambda)$; c) Studying algebraicity of values of hypergeometric functions at algebraic numbers can be regarded as special cases of the Zilber-Pink conjecture in Diophantine geometry. In this talk, I will explain how we use tools from mathematical logic to attack this conjecture. In particular, I will present some partial results toward the Zilber-Pink conjecture, including those proved by Christopher Daw and myself. |