It is known that a p-adic family of modular forms does not necessarily specialize into a classical modular form at weight 1, unlike the modular forms of weight 2 or higher. We will explain how this obstruction to classicality leads to a 'derived' action on modular forms of weight 1, which can be understood as the so-called derived Hecke operator at p. We will also investigate the role of the derived action in the study of p-adic periods of the adjoint of the weight 1 modular forms.
Let π be a cuspidal automorphic representation of Sp2n over ℚ which is holomorphic discrete series at infinity, and χ a Dirichlet character. Then one can attach to π an orthogonal p-adic Galois representation ρ of dimension 2n+1. Assume ρ is irreducible, that π is ordinary at p, and that p does not divide the conductor of χ. I will describe work in progress which aims to prove that the Bloch-Kato Selmer group attached to the twist of ρ by χ vanishes, under some mild ramification assumptions on π; this is what is predicted by the Bloch-Kato conjectures.
The proof uses "ramified Eisenstein congruences" by constructing p-adic families of Siegel cusp forms degenerating to Klingen Eisenstein series of non-classical weight, and using these families to construct ramified Galois cohomology classes for the Tate dual of the twist of ρ by χ.
Symmetric power functoriality is one of the basic cases of Langlands's functoriality conjectures and is the route to the proof of the Sato-Tate conjecture (concerning the distribution of the modulo p point counts of an elliptic curve over ℚ, as the prime p varies). I will discuss the proof of the existence of the symmetric power liftings of Hilbert modular forms of regular weight. The proof uses automorphy lifting theorems, automorphic forms on unitary groups, and the geometry of Shimura varieties, as well as the fact that Spec(ℤ) is simply connected.
In a recent machine learning based study, He, Lee, Oliver, and Pozdnyakov observed a striking oscillating pattern in the average value of the P-th Frobenius trace of elliptic curves of prescribed rank and conductor in an interval range. Sutherland discovered that this bias extends to Dirichlet coefficients of a much broader class of arithmetic L-functions when split by root number.
In my talk, I will discuss this root number correlation bias when the average is taken over weight 2 modular newforms of all Galois orbit sizes simultaneously. I will point to a source of this phenomenon in this case and compute the correlation function exactly.
We survey some recent developments in the theory of vector-valued modular forms for SL2(ℤ), focusing especially on our recent and ongoing joint work with Frank Calegari and Yunqing Tang that proved the Unbounded Denominators conjecture as one application.
The first talk will be an introduction to noncongruence modular forms, from one side, and from another side to arithmetic algebraization methods. We will discuss how to connect these two subjects, and the kind of further applications that arithmetic algebraization methods may have to offer in number theory. After the basic examples and some history, we will turn to Bost's slopes method of Arakelov theory for the technical underpinning of our proofs.
In the second talk, I will establish a new equivariant holonomy bound and apply it to prove the Unbounded Denominators conjecture of Atkin, Swinnerton-Dyer, and Mason. This will be a new argument alternative to our original proof in (F. Calegari, V. Dimitrov, Y. Tang: The unbounded denominators conjecture).
We survey some recent developments in the theory of vector-valued modular forms for SL2(ℤ), focusing especially on our recent and ongoing joint work with Frank Calegari and Yunqing Tang that proved the Unbounded Denominators conjecture as one application.
The first talk will be an introduction to noncongruence modular forms, from one side, and from another side to arithmetic algebraization methods. We will discuss how to connect these two subjects, and the kind of further applications that arithmetic algebraization methods may have to offer in number theory. After the basic examples and some history, we will turn to Bost's slopes method of Arakelov theory for the technical underpinning of our proofs.
In the second talk, I will establish a new equivariant holonomy bound and apply it to prove the Unbounded Denominators conjecture of Atkin, Swinnerton-Dyer, and Mason. This will be a new argument alternative to our original proof in (F. Calegari, V. Dimitrov, Y. Tang: The unbounded denominators conjecture).
Zeros of L-functions have been extensively studied, due to their close connection to arithmetic problems. Despite several precise conjectures about their behaviour, our unconditional understanding of them remains limited. In this talk we will discuss certain intrinsic properties of such zeros, focusing on what is known (in degrees 1 and 2) about their accumulation on the central line and their multiplicity. Here the tools of analytic number theory can give quantitative advances, and we will show how to deduce that there are many zeros of multiplicity one for the L-function associated to a modular form.
The behaviour of quadratic twists of modular L-functions is at the critical point is related both to coefficients of half integer weight modular forms and data on elliptic curves. Here we describe a proof of an asymptotic for the second moment of this family of L-functions, previously available conditionally on the Generalized Riemann hypothesis by the work of Soundararajan and Young. Our proof depends on deriving an optimal large sieve type bound.
A Diophantine upper bound on the dimensions of certain spaces of holonomic functions was the main ingredient in our proof with Calegari and Tang of the 'unbounded denominators conjecture' (presented by Tang in last year's number theory seminar) from the theory of non-congruence and vector-valued modular forms. In this talk, I will report on our sequel joint work-in-progress where we extend the scope of these arithmetic holonomy bounds to beneath the framework of finite-index subgroups of SL2(ℤ) and onto the arithmetic theory of certain periods appearing as Apery limits for local systems on the triply punctured projective line. Applications include irrationality proofs, with quantitative bad approximability measures, of the 2-adic realization of ζ(5), the archimedean period L(2,χ−3)−π(log3)/(3√3), and the products of two logarithms log(1−1/m)log(1−1/n) for arbitrary integer pairs n,m with 0<|1−m/n|<ϵ0, where ϵ0 is some positive absolute constant. As a by-product, we find an arithmetic characterization of the logarithm function.
Motivated by deep observations on elliptic curves/modular forms, Nicholas Katz proposed three problems on sign changes, equidistributions and modular structures of Kloosterman sums in 1980. In this talk, we will discuss some recent progresses towards these three problems made by analytic number theory (e.g., sieve methods and automorphic forms) combining certain tools from ℓ-adic cohomology.
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