Talks

We describe how Euler added up all positive integers into a mysterious fraction when he was 28 years old, and I try to legitimize his method “p-adically”. This is a story of Number Theory from the 17th century on. We only need some knowledge of polynomials and fractions of polynomials and very basics of differentiation. If time allows, I enter into some results related to Ramanujan I found when I was 28 years old. For the results exposed here, a detailed proof can be found in my book: “Elementary Theory of L-functions and Eisenstein Series,” LMSST vol. 26, 1993, Cambridge U. Press.

For a normalized newform g in S_{k}(\Gamma_{0}(N)) with complex multiplication by an imaginary quadratic field K, there is a mock modular form f^{+} corresponding to g. K. Bringmann, P. Guerzhoy, and B. Kane modified f^{+} to obtain the p-adic modular form by a certain p-adic constant \alpha_{g}. In addition, they showed that \alpha_{g}=0 if p is split in K and does not divide N. On the other hand, the speaker showed that \alpha_{g} is a p-adic unit for an inert prime p that does not divide 2N when \dim S_{k}(\Gamma_{0}(N))=1. In this talk, the speaker determines the p-adic valuation of \alpha_{g} for an inert prime p under a mild condition, when g has weight 2 and rational Fourier coefficients.

The celebrated BDP formula evaluates Rankin–Selberg p-adic L-functions at points outside their interpolation range in terms of Generalised Heegner cycles (a phenomenon referred to as wall-crossing). This principle has been extended to triple products by Bertolini–Seveso–Venerucci and Darmon–Rotger, who relate values of Hsieh’s unbalanced p-adic L-functions on the balanced range to diagonal cycles. I will report on a result where wall-crossing is used to factor a triple product p-adic L-function with an empty interpolation range, to yield a p-adic Artin formalism for families of the form f × g × g. The key input is the arithmetic Gan–Gross–Prasad (Gross–Kudla) conjecture, linking central derivatives of (complex) triple product L-functions to Bloch–Beilinson heights of diagonal cycles and their comparison with their GL(2) counterpart (Gross–Zagier formulae). I will also discuss an extension to families on GSp(4) × GL(2) × GL(2), where a new double wall-crossing phenomenon arises and is required to explain a p-adic Artin formalism for families of the form F x g x g. This suggests a higher BDP/arithmetic GGP formula concerning second-order derivatives.

I will present a construction of anticyclotomic Euler systems, for those Galois representations of a CM field that are conjugate-symplectic, automorphic, and of regular, "balanced" Hodge-Tate type. Its main ingredients are variants of the generating series of special cycles on unitary Shimura varieties studied by Kudla and Liu, and the construction is conditional on a conjecture on their modularity. The relevant notion of Euler system is the one studied by Jetchev-Nekovar-Skinner. Combining with their work and with a height formula obtained with Liu yields (unconditionally) some new cases of the p-adic Beilinson-Bloch-Kato conjecture in analytic rank one.

In this talk, I will survey recent developments about the connection between neural networks and models of quantum mechanics and quantum field theory. Previous work has shown that the neural network - Gaussian process correspondence can be interpreted as the statement that large-width neural networks share some properties with free, or weakly interacting, quantum field theories (QFTs). Here I will focus on 1d QFTs, or models of quantum mechanics, where one has greater theoretical control. For instance, under mild assumptions, one can prove that any model of a quantum particle admits a representation as a neural network. Cherished features of quantum mechanics, such as uncertainty relations, emerge from specific architectural choices that are made to satisfy the axioms of quantum theory. Based on 2504.05462 with Jim Halverson.

The pseudo-Dirac Higgsino is one of the last surviving electroweak WIMPs. The LHC will not reach the 1.1 TeV target mass even with full luminosity and prospects for its indirect detection depend on a favorable dark matter density profile at the galactic center. Since it has only off-diagonal couplings at tree-level, its direct detection is possible only when the mass splitting is smaller than the initial center of mass kinetic energy. This direct detection loophole is actually more generic; going by the name 'inelastic dark matter'. In this talk I will talk about my recent efforts to reach mass splittings larger than what has been thought to be possible thus far, by invoking interesting astrophysics, elements from the end of the periodic table, and large volume neutrino detectors. 

Euler Systems have proven to be versatile tools for understanding Selmer groups and their connections to special values of L-functions. However, despite the key role they have played in making progress toward foundational conjectures in number theory like the Birch–Swinnerton-Dyer and Bloch– Kato Conjectures, only a handful of provably non-trivial Euler systems have been constructed to date. A significant obstacle to constructing Euler Systems lies in producing candidate Galois cohomology classes. This lecture series presents a method to overcome this obstacle that does not rely on rare (known) motivic classes. We will focus on building ´etale cohomology classes originating from automorphic data: Eisenstein series and Theta series. This framework not only retrieves most classical Euler systems but can also be applied to construct an Euler system for the adjoint of an elliptic modular form.
References:
• C. Skinner, L-values and nonsplit extensions: a simple case, https://msp.org/ent/2024/3-1/p03.xhtml
• H. Darmon etal, p-adic L-functions and Euler systems: a tale in two trilogies.