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.

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