PI: Emmanuel Lecouturier

The BIMSA number theory group covers five broad areas in number theory: -Additive combinatorics (A. Best): Furstenberg-Sárközy and Szemerédi type theorems.- Arithmetic of $L$-functions: (W. He, E. Lecouturier, D. Yan, X. Yan, X. Zhu): Arithmetic statistics of Selmer groups, Iwasawa theory, BSD conjecture, Eisenstein congruences. - Analytic methods for $L$-functions and automorphic forms (C. Lupu, W. Wang, D. Wu): Multi zeta and $L$-functions, mixed polyharmonic automorphic forms, Riemann hypothesis. - $p$-adic geometry (T-A. Azzouz, Y-S. Moon, K. Shimizu): Families of $p$-adic Galois representations, $p$-adic Hodge theory, $p$-adic cohomologies, $p$-adic differential equations. -Automorphic forms (T. Deng): Stability results for Langlands Shahidi Gamma factors, Arthur packets for (quasi-plit) classical groups over R, Eisenstein cohomology. Yong-Suk Moon (2023). On Fontaine's conjecture for torsion crystalline local systems. (arXiv:2304.00855) We prove an analogue of Fontaine's conjecture for torsion crystalline local systems on the generic fiber of a smooth connected $p$-adic formal scheme, and show as an application that the locus of crystalline local systems whose Hodge-Tate weights lie in a fixed interval cuts out a closed subscheme of the universal deformation ring. Emmanuel Lecouturier (joint with Jun Wang, 2023). On the Birch and Swinnerton-Dyer conjecture for certain abelian varieties with a rational isogeny (arXiv:2305.00643). The Birch and Swinnerton-Dyer (BSD) conjecture predicts a relation between the analytic behaviour of the $L$-function of an abelian variety at $s=1$ and arithmetic invariants of that abelian variety. We prove a modulo $p$ version (a weak analogue) of the BSD conjecture in a very special case: for even quadratic twists of $p$-Eisenstein quotients of the Jacobian of $X_0(N)$ when $N$ is prime and $p\geq 5$ divides exactly $N-1$. Koji Shimizu (2022). A $p$-adic monodromy theorem for de Rham local systems. (Compositio Mathematica, 158(12), 2157-2205. doi:10.1112/S0010437X2200776X) The goal of $p$-adic Hodge theory is to understand $p$-adic Galois representations as well as their families parametrized by varieties ($p$-adic local systems). For this aim, Fontaine defined several classes of Galois representations such as de Rham and semistable representations. The article shows that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. Cezar Lupu (joint with Li Lai and Derek Orr, 2022). Elementary proofs of Zagier's formula for multiplezeta values and its odd variant. (arXiv:2201.09262) We prove the well-known Zagier's formula for multiple zeta values (MZV) involving Hoffman elements, $H(a, b)=\zeta(2, \ldots, 2, 3, 2, \ldots, 2)$ as a $\mathbb{Q}$-linear combination of powers of $\pi$ and odd zeta values. Moreover, a similar Zagier-type formula is obtained for multiple $t$-values (odd variant of MZVs). Dong Yan (2019). Stable lattices in modular Galois representations and Hida deformation. (J. Number Theory 197 (2019), 62–88, doi:0.1016/j.jnt.2018.06.014). In this paper we study the boundedness and unboundedness of the variation of the number of isomorphic classes of stable lattices in $p$-adic families of residually reducible ordinary modular Galois representations when the weight and the level vary. This turns out to prove the boundedness and unboundedness of the length of $\mu$-invariants of Selmer groups due to a formula of Perrin-Riou.