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We survey known results concerning cancellation laws in cardinal arithmetic without the axiom of choice. These results include the cancellation law for Dedekind finite cardinals in cardinal addition a...
In this talk, I will discuss arithmetic purity of strong approximation : motivations and related results. Finally, for complete toric varieties, I will briefly explain my work on this topic.
The Zariski dense orbit conjecture of the arithmetic dynamic is to study the existence of a rational point P with a Zariski dense orbit of a dominant rational self-map f of a projective variety X. Whe...
The classical Siegel–Weil formula relates theta series to Eisenstein series and its arithmetic version is central in Kudla's program. I will discuss arithmetic mixed Siegel-Weil formulas. I will focus...
Let E : 0 ! S ! E ! Q ! 0 be a short exact sequence of hermitian vector bundles with metrics on S and Q induced from that on E. We compute the Bott-Chern form áe(E ) corresponding to any characteri...
Let F be the complete °ag variety over SpecZ with the tautological ˉltration 0 ½ E1 ½ E2 ½ ¢ ¢ ¢ ½ En = E of the trivial bundle E of rank n over F. The trivial hermitian metric o...
Let G be the arithmetic Grassmannian over SpecZ with the natural invariant KÄahler metric on G(C). We study the combinatorics of the arithmetic Schubert calculus in the Arakelov Chow ring CH(G)
We prove the arithmetic Hodge index and hard Lefschetz conjectures for the Grassmannian G = G(2; N) parametrizing lines in projective space, for the natural arithmetic Lefschetz operator de ned via t...
Let S be a closed Shimura variety uniformized by the complex n-ball. The Hodge conjecture predicts that every Hodge class in H2k(S, Q), k = 0, . . . , n, is algebraic. We show that this holds for all ...
We show that special cycles generate a large part of the cohomology of locally symmetric spaces associated to orthogonal groups. We prove in particular that classes of totally geodesic submanifolds ge...
We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the sp...
In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ1-definable in (H(ω2), ...
We prove that the theta correspondence for the dual pair (SLg2; P B£), for B an indeˉnite quaternion algebra over Q, acting on modular forms of odd square-free level, preserves rationality and p-int...
Deligne and Rapoport developed the theory of generalized elliptic curves over arbitrary schemes and they proved that various moduli stacks for (ample) “level-N” structures on generalized elliptic cu...
An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we char...

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