Tag - Analytic number theory

Chao Li: On the Kudla-Rapoport conjecture

9th April, 2020

The Kudla-Rapoport conjecture predicts a precise identity between the arithmetic intersection number of special cycles on unitary Rapoport-Zink spaces and the derivative of local representation densities of hermitian forms. It is a key local ingredient to establish the arithmetic Siegel-Weil formula and the arithmetic Rallis inner product formula, relating the height of special cycles on Shimura varieties to the derivative of Siegel Eisenstein series and L-functions. We will motivate this conjecture, explain a proof and discuss global applications.

Mikolaj Fraczyk: Density conjecture for horizontal families of lattices in SL(2)

2nd April, 2020

Let G be a real semisimple Lie group with an irreducible unitary representation π. The non-temperedness of π is measured by the parameter p(π), which is defined as the infimum of p≥2 such that π has matrix coefficients in Lp(G). Sarnak and Xue conjectured that for any arithmetic lattice Γ⊂G and principal congruence subgroup Γ(q)⊂Γ, the multiplicity of π in L2(G/Γ(q)) is at most O(V(q)2/p(π)+ε) where V(q) is the covolume of Γ(q). In some contexts such estimate is a decent substitute for the Ramanujan conjecture. For G of real rank 1 Sarnak and Xue translate the estimate into a Diophantine counting problem which they managed to solve for SL2(ℝ) and SL2(ℂ).

In this talk I will explain how one can get the same multiplicity bounds for families of pairwise non-commensurable lattices in G=SL2(ℝ), SL2(ℂ) given as unit groups of maximal orders of quaternion algebras over number fields (“horizontal families”). Namely: m(π,Γ)≪V2/p(π)+ε, where V is the covolume of Γ. I will also discuss similar bounds on multiplicities of representations π1×π2 of G=SL2(ℝ)2 where π1 is fixed non-tempered but π2 is allowed to vary together with the lattice.

Paul Nelson: Eisenstein series and the cubic moment for PGL2

28th January, 2020

We will discuss how to study the cubic moment of any family of automorphic L-functions on PGL2 using regularized diagonal periods of Eisenstein series, following a strategy suggested by Michel-Venkatesh. Applications include generalizations to the setting of number fields of some results of Conrey-Iwaniec and Petrow-Young, improved estimates for representation numbers of ternary quadratic forms over number fields, and improvements to the prime geodesic theorem on arithmetic hyperbolic 3-folds.

Peter Sarnak: Integer Points on Affine Cubic Surfaces

1st November, 2019

The level sets of a cubic polynomial in four or more variables tend to have many integer solutions, while ones in two variables a limited number of solutions. Very little is known in the case of three variables. For cubics which are character varieties (thus carrying a non-linear group of morphisms) a Diophantine analysis has been developed and we will describe it. Passing from solutions in integers to integers in, say, a real quadratic field there is a fundamental change which is closely connected to challenging questions about one-commutators in 𝔰𝔩2 over such rings.

Trevor Wooley: A slice or two of a diagonal cubic: arithmetic stratification via the circle method

24th October, 2019

Some 25 years ago, as announced at a previous version of this seminar, Vaughan and the speaker obtained asymptotic upper and lower bounds for the number of non-trivial integral points on the Segre cubic

x13+...+x63=x1+...+x6=0

with naive height bounded by a large parameter B. Seeking an explanation for the "unexpected" growth rate B2(log B)5, they offered a heuristic explanation for the role of the major and minor arc contributions in the application of the circle method to this problem, and connected these terms with Manin’s ideas on arithmetic stratification. We now consider diagonal cubics in more variables with one or two linear slices in the light of recent progress on Vinogradov’s mean value theorem. In particular, in work joint with Joerg Bruedern, we are able to prove an asymptotic formula in which Manin’s arithmetic stratification identifies naturally and provably in terms of major and minor arc contributions from the circle method.

Peter Sarnak: Applications of Points on Subvarieties of Tori

1st October, 2019

The intersection of the division group of a finitely generated subgroup of a torus with an algebraic sub-variety has been understood for some time (Lang, Laurent). After a brief review of some of the tools in the analysis and their recent extensions (André-Oort conjectures), we give some old and new applications; periodicity of Betti numbers, algebraicity of Painlevé equations, and the additive structure of the spectra of quantum graphs.

Raphael Steiner: Taking the Hecke algebra to its limits

12th September, 2019

We parametrize elements in the full Hecke algebra in a way such that the parametrization represents a generic automorphic form. By convolving, we then arrive at pre-trace formulas which are modular in three variables. From here, various identities for higher moments may be derived. We give applications to the sup-norm and fourth-norm of holomorphic Hecke eigenforms as well as Hecke-Maass forms on Γ \ ℍ and furthermore outline future work on higher moments of periods and quantum variance. This is joint work with Ilya Khayutin.

Jan Vonk: Singular moduli for real quadratic fields

4th April, 2019

The theory of complex multiplication describes finite abelian extensions of imaginary quadratic number fields using singular moduli, which are special values of modular functions at CM points. I will describe joint work with Henri Darmon in the setting of real quadratic fields, where we construct p-adic analogues of singular moduli through classes of rigid meromorphic cocycles. I will discuss p-adic counterparts for our proposed RM invariants of classical relations between singular moduli and analytic families of Eisenstein series.

Yiannis Sakellaridis: Beyond Endoscopy: Local aspects of Venkatesh’s thesis

14th March, 2019

The thesis of Akshay Venkatesh obtains a "Beyond Endoscopy" proof of stable functorial transfer from tori to SL2, by means of the Kuznetsov formula. In this talk, I will show that there is a local statement that underlies this work; namely, there is a local transfer operator taking orbital measures for the Kuznetsov formula to test measures on the torus. The global comparison of trace formulas is then obtained as a Poisson summation formula for this transfer operator.

Emmanuel Lecouturier: An Application of a Conjecture of Mazur-Tate to Supersingular Elliptic Curves

14th February, 2019

In 1987, Barry Mazur and John Tate formulated refined conjectures of the "Birch and Swinnerton-Dyer type", and one of these conjectures was essentially proved in the prime conductor case by Ehud de Shalit in 1995. One of the main objects in de Shalit's work is the so-called refined L-invariant, which happens to be a Hecke operator. We apply some results of the theory of Mazur's Eisenstein ideal to study in which power of the Eisenstein ideal L belongs. As a corollary of our study, we give a surprising elementary formula on supersingular j-invariants.