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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.

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.

The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.

A bit of representation theory of groups over local fields, parabolic induction, cuspidal representations. Review of the Local Langlands correspondence, L-functions and epsilon factors. L-packets, the Jacquet-Langlands correspondence, The GGP conjectures: both local and global conjectures.

We first prove, for a prime p>3 unramified in a CM quadratic extension of a totally real field F, h(M/F)L(\chi)|H(\psi)|h(M/F)F(\chi) (h(M/F)=h(M)/h(F)) in \Lambda for the congruence power serie H(\psi) of \psi lifting a fixed anti-cyclotomic character \chi and anticyclotomic Katz p-adic L-function L(\chi) of branch character \chi, built on the lectures by Tilouine proving this over \Lambda[1/p]. Here \Lambda is the many variable Iwasawa algebra of M. In the second lecture, we give a sketch of the proof of the reverse divisibility: H(\psi)|h(M/F)L(\chi) resulting in the main conjecture, as H(\psi)=h(M/F)F(\chi) for the anticyclotomic Iwasawa power series F(\chi) by the “R=T”-theorem.

The arithmetic Siegel-Weil formula, conjectured by Kudla-Rapoport and proved by Li-Zhang, expresses the degrees of certain 0-cycles on integral models of unitary Shimura varieties in terms of the nondegenerate Fourier coefficients of the central derivative of an Eisenstein series.
Feng-Yun-Zhang proved a higher derivative version of this arithmetic Siegel-Weil formula in the function field setting, now expressing degrees of 0-cycles on moduli spaces of unitary shtukas to the nondegenerate Fourier coefficients of higher central derivatives of an Eisenstein series.
The goal of my lecture series is (1) to explain all of this background, (2) extend the results of Feng-Yun-Zhang to include some degenerate coefficients, and (3) deduce from this extension an arithmetic application: the nonvanishing of higher central derivatives of certain Langlands L-functions implies the nonvanishing of classes in the Chow groups of moduli spaces of shtukas.
All of the new results are joint work with Tony Feng and Mikayel Mkrtchyan.

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.