On the p-adic Beilinson conjecture and the equivariant Tamagawa number conjecture

Let E/K be a finite Galois extension of totally real number fields with Galois group G. Let p be an odd prime and let $$r>1$$ r > 1 be an odd integer. The p-adic Beilinson conjecture relates the values at $$s=r$$ s = r of p-adic Artin L-functions attached to the irreducible characters of G to those of corresponding complex Artin L-functions. We show that this conjecture, the equivariant Iwasawa main conjecture and a conjecture of Schneider imply the ‘p-part’ of the equivariant Tamagawa number conjecture for the pair $$(h^0(\mathrm {Spec}(E))(r), \mathbb {Z}[G])$$ ( h 0 ( Spec ( E ) ) ( r ) , Z [ G ] ) . If $$r>1$$ r > 1 is even we obtain a similar result for Galois CM-extensions after restriction to ‘minus parts’.

THE ZEROTH -STABLE HOMOTOPY SHEAF OF A MOTIVIC SPACE

We establish a kind of ‘degree $0$ Freudenthal ${\mathbb {G}_m}$ -suspension theorem’ in motivic homotopy theory. From this we deduce results about the conservativity of the $\mathbb P^1$ -stabilization functor. In order to establish these results, we show how to compute certain pullbacks in the cohomology of a strictly homotopy-invariant sheaf in terms of the Rost–Schmid complex. This establishes the main conjecture of [2], which easily implies the aforementioned results.

On two conjectures of Gross for large vanishing orders

We prove the Gross order of vanishing conjecture in special cases where the vanishing order of the character in question can be arbitrarily large. In almost all previously known cases the vanishing order is zero or one. One major ingredient of our proofs is the equivalence of this conjecture to the Gross–Kuz’min conjecture. We present here a direct proof of this equivalence, using only the known validity of the Iwasawa Main Conjecture over totally real fields.

Characteristic cycle and wild ramification for nearby cycles of étale sheaves

In this article, we give a bound for the wild ramification of the monodromy action on the nearby cycles complex of a locally constant étale sheaf on the generic fiber of a smooth scheme over an equal characteristic trait in terms of Abbes and Saito’s logarithmic ramification filtration. This provides a positive answer to the main conjecture in [24] for smooth morphisms in equal characteristic. We also study the ramification along vertical divisors of étale sheaves on relative curves and abelian schemes over a trait.

Arithmetic of higher-dimensional orbifolds and a mixed Waring problem

We study the density of rational points on a higher-dimensional orbifold $$(\mathbb {P}^{n-1},\Delta )$$ ( P n - 1 , Δ ) when $$\Delta $$ Δ is a $$\mathbb {Q}$$ Q -divisor involving hyperplanes. This allows us to address a question of Tanimoto about whether the set of rational points on such an orbifold constitutes a thin set. Our approach relies on the Hardy–Littlewood circle method to first study an asymptotic version of Waring’s problem for mixed powers. In doing so we make crucial use of the recent resolution of the main conjecture in Vinogradov’s mean value theorem, due to Bourgain–Demeter–Guth and Wooley.

Supersaturation, Counting, and Randomness in Forbidden Subposet Problems

In the area of forbidden subposet problems we look for the largest possible size $La(n,P)$ of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain a forbidden inclusion pattern described by $P$. The main conjecture of the area states that for any finite poset $P$ there exists an integer $e(P)$ such that $La(n,P)=(e(P)+o(1))\binom{n}{\lfloor n/2\rfloor}$. In this paper, we formulate three strengthenings of this conjecture and prove them for some specific classes of posets. (The parameters $x(P)$ and $d(P)$ are defined in the paper.) For any finite connected poset $P$ and $\varepsilon>0$, there exists $\delta>0$ and an integer $x(P)$ such that for any $n$ large enough, and $\mathcal{F}\subseteq 2^{[n]}$ of size $(e(P)+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$, $\mathcal{F}$ contains at least $\delta n^{x(P)}\binom{n}{\lfloor n/2\rfloor}$ copies of $P$. The number of $P$-free families in $2^{[n]}$ is $2^{(e(P)+o(1))\binom{n}{\lfloor n/2\rfloor}}$. Let $\mathcal{P}(n,p)$ be the random subfamily of $2^{[n]}$ such that every $F\in 2^{[n]}$ belongs to $\mathcal{P}(n,p)$ with probability $p$ independently of all other subsets $F'\in 2^{[n]}$. For any finite poset $P$, there exists a positive rational $d(P)$ such that if $p=\omega(n^{-d(P)})$, then the size of the largest $P$-free family in $\mathcal{P}(n,p)$ is $(e(P)+o(1))p\binom{n}{\lfloor n/2\rfloor}$ with high probability.