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Emmanuel Lecouturier: On the BSD conjecture for certain families of abelian varieties with rational torsion

24th March, 2022

Let N and p at least 5 be primes such that p divides N−1. In his landmark paper on the Eisenstein ideal, Mazur proved the p-part of the BSD conjecture for the p-Eisenstein quotient J(p) of J0(N) over ℚ. Using recent results and techniques of the work of Venkatesh and Sharifi on the Sharifi conjecture, we prove unconditionally a weak form of the BSD conjecture for J(p) over a quadratic field K (which can be real or imaginary). This includes results in positive analytic rank, as the analytic rank of J(p) over K can be greater than or equal to 2 for well-chosen K.

Chandrashekhar Khare: A Wiles-Diamond numerical criterion in higher dimensions

17th February, 2022

Wiles's proof of the modularity of (semistable) elliptic curves over the rationals and Fermat’s Last Theorem relied on his invention of a modularity lifting method. There were two strands to the method:

   1.  A numerical criterion to for a map of rings to be an isomorphism between complete intersections that are finite flat over ℤp in Wiles's paper on FLT, subsequently generalized by Fred Diamond.
   2.  Patching (in his paper with Taylor)

The patching method has been vastly generalized; in particular Calegari-Geraghty found a way to generalize it in principle to prove (potential) modularity of elliptic curves over imaginary quadratic fields (a situation of "positive defect"). Their method has been made unconditional to prove modularity lifting results over CM fields in the ten author paper. The numerical criterion has yet to be generalized to positive defect.

In joint work with Srikanth Iyengar and Jeff Manning we give a development of the Wiles-Diamond numerical criterion to situations of positive defect (for example to proving modularity results for torsion Galois representations over imaginary quadratic fields). This in principle allows one to prove integral R=T theorems (in minimal and non-minimal situations), for which just the use of patching seems inadequate. One interest of proving such integral versions of modularity lifting is that in these situations, the Betti cohomology groups of 3-dimensional Bianchi manifolds (the analog of the modular curves over imaginary quadratic fields) have a lot of torsion. Our strategy consists of proving a higher dimensional version of the numerical criterion of Wiles-Diamond and applying it to prove integral R=T theorems (in the non-minimal case) after patching.

Chandrashekhar Khare: Modular forms, Galois representations and the Ramanujan prime 691

2nd February, 2022

The number 1729 is part of the Ramanujan lore, being famously the number of the taxi which Hardy took to visit Ramanujan. In this talk I would like to argue that the prime number 691 is a number of greater significance than 1729 to the mathematics that Ramanujan discovered. It occurs in his paper On certain arithmetical functions published in 1916 in which he observed a congruence between modular forms modulo 691. The observations Ramanujan made in that paper had a great influence on the developments in number theory in the 20th century which led to a proof of Fermat's Last Theorem by Andrew Wiles. The 1916 observations of Ramanujan led to Serre formulating his influential modularity conjecture which was proved in 2009 by Jean-Pierre Wintenberger and myself. I will explain some of these developments and current work on the connection between modular forms and Galois representations.

V Kumar Murty: ζ(3), log 2 and π

11th January, 2022

Values of the Riemann zeta function at odd positive integers have proved enigmatic over several centuries of study. In 1740, Euler asked whether ζ(3) could be expressed algebraically in terms of log 2 and π. In this talk, we shall show that the Grothendieck period conjecture applied to certain mixed motives answers Euler's question in the negative.

Victor Wang: Conditional approaches to sums of cubes

18th November, 2021

In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number of integral solutions to x13+...+x63 in expanding boxes.

Building on Hooley's work, I will sketch new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families - e.g. conditional statistical results on sums of three integer cubes (a project suggested by Amit Ghosh and Peter Sarnak). These form the bulk of my thesis work (advised by Sarnak), and involve phenomena both random and structured, average- and worst-case, and multiplicative and additive.

Yunqing Tang: The unbounded denominators conjecture

11th November, 2021

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite-index subgroup of SL2(ℤ) whose Fourier coefficients have bounded denominators must be a modular form for some congruence subgroup. In this talk, we will give a sketch of the proof of this conjecture based on a new arithmetic algebraization theorem. This is joint work with Frank Calegari and Vesselin Dimitrov.

Paul Nelson: Bounds for standard L-functions

7th October, 2021

We consider the standard L-function attached to a cuspidal automorphic representation of a general linear group. We present a proof of a subconvex bound in the t-aspect. More generally, we address the spectral aspect in the case of uniform parameter growth.

Will Sawin: Sums in progressions over 𝔽q[T], the symmetric group, and geometry

30th September, 2021

I will discuss some recent progress in analytic number theory for polynomials over finite fields, giving strong new estimates for the number of primes in arithmetic progressions, as well as for sums of some arithmetic functions in arithmetic progressions. The strategy of proof is fundamentally geometric, and I will explain some of the geometric ideas in the proof, including how we can use the representation of the symmetric group to handle many different arithmetic functions in a uniform way.

Maksym Radziwill: Expansion and parity

13th May, 2021

I will discuss recent work with Harald Helfgott in which we establish roughly speaking that the graph connecting n to n ± p with p a prime dividing n is almost "locally Ramanujan". As a result we obtain improvements of results of Tao and Tao-Teravainen on logarithmic Chowla. I will discuss the main ideas in the proof and the connections with logarithmic Chowla.