Séminaire de théorie des nombres de l'IMJ-PRG

For $n>1$, consider an absolutely irreducible polynomial $F(Y,X_1,...,X_n)$ that is a polynomial in $Y^m$ and monic in $Y$. Let $N(F,B)$ be the number of integral vectors $x$ of height at most $B$ such that there is an integral solution to $F(Y,x)=0$. For $m>1$ unconditionally, and $m=1$ under GRH, we show that $N(F,B) \ll_{\epsilon} log(||F||) ^c B^{n-1+1/(n+1)+\epsilon}$ under a non-degeneracy condition that encapsulates that $F(Y,X_1,...,X_n)$ is truly a polynomial in $n+1$ variables. A strength of this result is that it requires no smoothness assumptions for $F(Y,X_1,...,X_n)$ nor constraints on the degrees of $F$ in $X_1,...,X_n$. A key ingredient in this work is a formulation of the Katz-Laumon stratification theorems for exponential sums that is uniform in families. This talk is based on joint work with Dante Bonolis, Emmanuel Kowalski, and Lillian B. Pierce.

We will present the definition of Hyodo-Kato stacks, which intuitively are de Rham stacks of relative Fargues--Fontaine curves. After explaining their basic properties, we will present a new proof of the $p$-adic monodromy theorem, which avoids any deeper analysis of $p$-adic differential equations. This talk is based on joint work with Bosco/Le Bras/Rodriguez-Camargo/Scholze.

We prove a classification of analytic $p$-divisible groups over perfectoid spaces $S$ over $\mathbb{Q}_p$ in terms of Hodge--Tate triples on $S$, generalizing a theorem of Fargues. From this, we construct an analytic Dieudonné theory with values in mixed characteristic Shtukas over the Fargues--Fontaine disc. We use our results to realize the local Shimura varieties of EL and PEL type of Scholze--Weinstein as moduli spaces of analytic $p$-divisible groups, and we reinterpret the Hodge--Tate period map of Scholze in terms of $p$-topological torsion subgroups of abelian varieties.

Iwasawa theory of CM elliptic curves is well developed for primes $p$ split in the CM field (good ordinary case), and has applications to the BSD conjecture. In contrast, for $p$ inert (good supersingular) or ramified (bad additive), new phenomena occur and the theory is still fragmentary. For the anticyclotomic deformation at an inert prime, the last few years has seen progress due to the work of Kobayashi-Ota- , following Rubin's pioneering work in the mid 80's. In this talk, we report on a similar progress for ramified primes (joint with S. Kobayashi, K. Nakamura and K. Ota).

The theory of complex multiplication implies that the values of modular functions at CM points belong to abelian extensions of imaginary quadratic fields. In this talk, we propose a conjectural extension of this phenomenon to the setting of totally real fields. Generalizing the work of Darmon, Pozzi, and Vonk, we construct rigid cocycles for $\textrm{SL}(n)$, which play the role of modular functions, and define their values at points associated with totally real fields. The construction of these cocycles originates from a topological source: the Eisenstein class of a torus bundle. This is ongoing joint work with Peter Xu.

The Catalan constant is the alternating sum of the reciprocals of the squares of odd positive integers ($1-1/9+1/25-\cdots$). In this talk, we will describe a geometric construction of a 2-dimensional mixed motive over the field of rational numbers that has the Catalan constant as a period. We will use this motive to obtain a supply of linear forms in 1 and the Catalan constant. We will also see how the coefficients of 1 and the Catalan constant in these linear forms can explicitly be calculated. The talk is based on a joint work with Kumar Murty and Yusuke Nemoto.

TBA