Tag - Analytic number theory

Pierre Debes: The Hilbert-Schinzel specialization property

10th January, 2021

Hilbert's Irreducibility Theorem shows that irreducibility over the field of rationals is 'often' preserved when one specializes a variable in some irreducible polynomial in several variables. I will present a version 'over the ring' for which the specialized polynomial remains irreducible over the ring of integers. The result also relates to the Schinzel Hypothesis about primes in value sets of polynomials: I will discuss a weaker 'relative' version for the integers and the full version for polynomials. The results extend to other base rings than the ring of integers; the general context is that of rings with a product formula.

Joni Teräväinen: On the Liouville function at polynomial arguments

10th December, 2020

Let λ be the Liouville function and P(x) any polynomial that is not a square. An open problem formulated by Chowla and others asks to show that the sequence λ(P(n)) changes sign infinitely often. We present a solution to this problem for new classes of polynomials P, including any product of linear factors or any product of quadratic factors of a certain type. The proofs also establish some nontrivial cancellation in Chowla and Elliott type correlation averages.

Jingwei Xiao: A Unitary Analogue of Friedberg-Jacquet Periods and Central Values of Standard L Functions on GL(2n)

3rd December, 2020

Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(AF). They are often related to special values of certain L-functions. One of the most notable cases is when (G,H)=(U(n+1)☓U(n), U(n)), and these periods are related to central values of Rankin-Selberg L-functions on GL(n+1)☓GL(n). In this talk, I will explain my work in progress with Wei Zhang that studies central values of standard L-functions on GL(2n) using (G,H)=(U(2n), U(n)☓U(n)) and some variants. I shall explain the conjecture and a relative trace formula approach to study it. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma and Hironaka’s characterization of spherical functions on the space of non-degenerate Hermitian matrices. Also, the question admits an arithmetic analogue.

Shai Evra: Ramanujan Conjecture and the Density Hypothesis

19th November, 2020

The Generalized Ramanujan Conjecture (GRC) for GLn is a central open problem in modern number theory. Its resolution is known to yield several important applications. For instance, the Ramanujan-Petersson conjecture for GL2, proven by Deligne, was a key ingredient in the work of Lubotzky-Phillips-Sarnak on Ramanujan graphs.

One can also state analogues of (Naive) Ramanujan Conjectures (NRC) for other reductive groups. However, in the 70s Kurokawa and Howe-Piatetski-Shapiro proved that the (NRC) fails even for quasi-split classical groups.

In the 90s Sarnak-Xue put forth a Density Hypothesis version of the (NRC), which serves as a replacement of the (NRC) in applications.

In this talk I will describe a possible approach to proving the Density Hypothesis for definite classical groups, by invoking deep and recent results coming from the Langlands programme: The endoscopic classification of automorphic representations of classical groups due to Arthur, and the proof of the Generalized Ramanujan-Petersson Conjecture.

Levent Alpoge: Effective height bounds for odd-degree totally real points on some curves

12th November, 2020

Let o be an order in a totally real field, say F. Let K be an odd-degree totally real field. Let S be a finite set of places of K. We study S-integral K-points on integral models Ho of Hilbert modular varieties because not only do said varieties admit complete curves (thus reducing questions about such curves' K-rational points to questions about S-integral K-points on these integral models), they also have their S-integral K-points controlled by known cases of modularity, in the following way. First assume for clarity modularity of all GL2-type abelian varieties over K: then all S-integral K-points on Ho arise from K-isogeny factors of the [F:ℚ]-th power of the Jacobian of a single Shimura curve with level structure (by Jacquet-Langlands transfer). By a generalization of an argument of von Känel, isogeny estimates of Raynaud/Masser-Wüstholz and Bost's lower bound on the Faltings height suffice to then bound the heights of all points in Ho(oK,S). As for the assumption, though modularity is of course not known in this generality, by following Taylor's (sufficiently explicit for us) proof of his potential modularity theorem we are able to make the above unconditional.

Finally we use the hypergeometric abelian varieties associated to the arithmetic triangle group Δ(3,6,6) to give explicit examples of curves to which the above height bounds apply. Specifically, we prove that, for a∈ ℚ̅ x totally real of odd degree (e.g. a = 1) and L/ℚ(a) totally real of odd degree, there is an effectively computable c = ca,L∈ ℤ+ such that all x,y∈L satisfying x6 + 4y3 = a2 satisfy h(x) < c. Note that this gives infinitely many curves for each of which Faltings' theorem is now effective over infinitely many number fields.

Tasho Kaletha: An explicit supercuspidal local Langlands correspondence

29th October, 2020

We will give an explicit construction and description of a supercuspidal local Langlands correspondence for any p-adic group G that splits over a tame extension, provided p does not divide the order of the Weyl group. This construction matches any discrete Langlands parameters with trivial monodromy to an L-packet consisting of supercuspidal representations, and describes the internal structure of these L-packets.

The construction has two parts. The depth-zero part involves generalizing to disconnected groups results of Lusztig on the decomposition of a non-singular Deligne-Lusztig induction. Higher multiplicities occur in this decomposition and are handled using work of Bonnafé-Dat-Rouquier. The positive-depth part involves functorial transfer from a twisted Levi subgroup, which is made possible by an improvement of Yu's construction of supercuspidal representations obtained in recent joint work with Fintzen and Spice, and consideration of Harish Chandra characters.

We will also discuss ongoing work towards related conjectures: Shahidi's generic L-packet conjecture, Hiraga-Ichino-Ikeda formal degree conjecture,  stability and endoscopic transfer.

Jessica Fintzen: Representations of p-adic groups and applications

8th October, 2020

The Langlands programme is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress. I will also outline how new results about the representation theory of p-adic groups can be used to obtain congruences between arbitrary automorphic forms and automorphic forms which are supercuspidal at p, which is joint work with Sug Woo Shin. This simplifies earlier constructions of attaching Galois representations to automorphic representations, i.e. the global Langlands correspondence, for general linear groups. Moreover, our results apply to general p-adic groups and have therefore the potential to become widely applicable beyond the case of the general linear group.

Alexandra Florea: Non-vanishing of cubic L-functions over function fields

21st September, 2020

Chowla conjectured that L(1/2,𝝌) never vanishes, for 𝝌 any Dirichlet character. Soundararajan showed that more than 87.5\% of the values L(1/2,𝝌d), for 𝝌d a quadratic character, do not vanish. Much less is known about cubic characters. Baier and Young showed that more than X6/7-𝜺 of L(1/2,𝝌) are non-vanishing, for 𝝌 a primitive, cubic character of conductor of size up to X. I will talk about recent joint work with C. David and M. Lalin, where we show that a positive proportion of these central L-values are non-vanishing in the function field setting. This is achieved by computing the first mollified moment using techniques previously developed by the authors in their work on the first moment of cubic L–functions, and by obtaining a sharp upper bound for the second mollified moment, building on work of Soundararajan, Harper and Lester–Radziwill.

Richard Hatton: Heegner points and self-points on elliptic curves

21st September, 2020

In the arithmetic of elliptic curves, we are interested in the construction of points on an elliptic curve. In particular, it has been shown that we are able to bound certain Selmer groups using modular points, specifically the use of Heegner points by Kolyvagin and self points by Wuthrich. We will define these points and will show how they can be used to create the bounds and its generalisations.