The Bloch-Kato conjecture, relating special values of L-functions to algebraic data, is one of the most important open problems in number theory; it includes the Birch-Swinnerton-Dyer conjecture for elliptic curves as a special case. I will describe some recent breakthroughs establishing special cases of this conjecture (and related problems such as the Iwasawa
main conjecture) using the method of Euler systems.
We ask the question, "how does the infinite q-Pochhammer symbol transform under modular transformations?" and connect the answer to that question to the Stark conjectures. The infinite q-Pochhammer symbol transforms by a generalized factor of automorphy, or modular 1-cocycle, that is analytic on a cut complex plane. This "Shintani–Faddeev modular cocycle" is an SL2(ℤ)-parametrized family of functions generalizing Shintani’s double sine function and Faddeev’s non-compact quantum dilogarithm. We relate real multiplication values of the Shintani-Faddeev modular cocycle to exponentials of certain derivative L-values, conjectured by Stark to be algebraic units generating abelian extensions of real quadratic fields.
The braid group B2g+1 has a description in terms of the hyperelliptic mapping class group of a curve X of genus g. This equips it with an action on V = H1(X), and we may produce a wealth of new representations Sλ(V) by applying Schur functors to V. The goal of this talk is to describe the stable (in g) group homology of these representations. Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space. We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces. This is joint work with Bergström, Diaconu, and Petersen. Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.
This is a report of joint work with Bergström-Diaconu-Westerland and Miller-Patzt-Randal-Williams. Based on random matrix theory, Conrey-Farmer-Keating-Rubinstein-Snaith have conjectured precise asymptotics for moments of families of quadratic L-functions over number fields. There is an extremely similar function field analogue, worked out by Andrade-Keating. I will explain that one can relate this problem to understanding the homology of the braid group with certain symplectic coefficients. With Bergström-Diaconu-Westerland we compute the stable homology groups of the braid groups with these coefficients, together with their structure as Galois representations. (This will be explained in Craig Westerland's lecture on Nov 2.) We moreover show that the answer matches the number-theoretic predictions. With Miller-Patzt-Randal-Williams we prove an improved range for homological stability with these coefficients. (This will be explained in my lecture on Nov 3.) Together, these results imply the conjectured asymptotics for all moments in the function field case, for all sufficiently large (but fixed) q.
L-functions of degree d can be parametrized, in two different ways, by points with an attached multiplicity in (d-1)-dimensional Euclidean space. One approach separates the L-functions according to the shape of the Gamma-factors in the functional equation, equivalently, according to the infinity type of the underlying automorphic representation. The other approach combines all the L-functions of a given degree into a single picture in which the points, to leading order, are uniformly dense. We will describe these classifications and provide examples of several 'landscapes' in the L-function world.
A conjecture of Chowla postulates that no L-function of Dirichlet characters over the rationals vanishes at s=1/2. Soundararajan has proved non-vanishing for a positive proportion of quadratic characters. Over function fields Li has discovered that Chowla's conjecture fails for infinitely many distinct quadratic characters. However, on the basis of the Katz-Sarnak heuristics, it is still widely believed that one should have non-vanishing for 100% of the characters in natural families (such as the family of quadratic characters). Works of Bui-Florea, David-Florea-Lalin, Ellenberg-Li-Shusterman, among others, provided evidence giving a positive proportion of non-vanishing in several such families. I will present an upcoming joint work with Peter Koymans and Mark Shusterman, where we prove that for each fixed q congruent to 3 modulo 4 one has 100% non-vanishing in the family of imaginary quadratic function fields.
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).
Random matrix theory is a powerful tool for prediction in analytic number theory. Through this random matrix analogy, Fyodorov, Hiary and Keating conjectured very precisely the typical values of the Riemann zeta function in short intervals of the critical line, in particular their maximum. Their prediction relied on techniques from statistical mechanics such as the replica method, giving extreme values in disordered systems. Recent rigorous progress has exploited underlying branching structures instead, both for random characteristic polynomials and L-functions.
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