Arithmetic Intersection Research Articles

Dynamics on ℙ1: preperiodic points and pairwise stability

DeMarco, Krieger, and Ye conjectured that there is a uniform bound B, depending only on the degree d, so that any pair of holomorphic maps $f, g :{\mathbb {P}}^1\to {\mathbb {P}}^1$ with degree $d$ will either share all of their preperiodic points or have at most $B$ in common. Here we show that this uniform bound holds for a Zariski open and dense set in the space of all pairs, $\mathrm {Rat}_d \times \mathrm {Rat}_d$ , for each degree $d\geq 2$ . The proof involves a combination of arithmetic intersection theory and complex-dynamical results, especially as developed recently by Gauthier and Vigny, Yuan and Zhang, and Mavraki and Schmidt. In addition, we present alternate proofs of the main results of DeMarco, Krieger, and Ye [Uniform Manin-Mumford for a family of genus 2 curves, Ann. of Math. (2) 191 (2020), 949–1001; Common preperiodic points for quadratic polynomials, J. Mod. Dyn. 18 (2022), 363–413] and of Poineau [Dynamique analytique sur $\mathbb {Z}$ II : Écart uniforme entre Lattès et conjecture de Bogomolov-Fu-Tschinkel, Preprint (2022), arXiv:2207.01574 [math.NT]]. In fact, we prove a generalization of a conjecture of Bogomolov, Fu, and Tschinkel in a mixed setting of dynamical systems and elliptic curves.

On Local Representation Densities of Hermitian Forms and Special Cycles

Abstract In this paper, we reformulate conjectural formulas for the arithmetic intersection numbers of special cycles on unitary Shimura varieties with minuscule parahoric level structure in terms of weighted counting of lattices containing special homomorphisms.

An intersection formula for CM cycles on Lubin–Tate spaces

We give an explicit formula for the arithmetic intersection number of complex multiplication (CM) cycles on Lubin–Tate spaces for all levels. We prove our formula by formulating the intersection number on the infinite level. Our CM cycles are constructed by choosing two separable quadratic extensions K1,K2 over a non-Archimedean local field F. Our formula works for all cases: K1 and K2 can be either the same or different, ramified or unramified over F. This formula translates the linear arithmetic fundamental lemma (linear AFL) into a comparison of integrals. As an example, we prove the linear AFL for GL2(F).

On the arithmetic Siegel–Weil formula for GSpin Shimura varieties

We formulate and prove a local arithmetic Siegel–Weil formula for GSpin Rapoport–Zink spaces, which is a precise identity between the arithmetic intersection numbers of special cycles on GSpin Rapoport–Zink spaces and the derivatives of local representation densities of quadratic forms. As a first application, we prove a semi-global arithmetic Siegel–Weil formula as conjectured by Kudla, which relates the arithmetic intersection numbers of special cycles on GSpin Shimura varieties at a place of good reduction and the central derivatives of nonsingular Fourier coefficients of incoherent Siegel Eisenstein series.

Numerical equivalence of ℝ-divisors and Shioda–Tate formula for arithmetic varieties

Abstract Let X be an arithmetic variety over the ring of integers of a number field K, with smooth generic fiber X K {X_{K}} . We give a formula that relates the dimension of the first Arakelov–Chow vector space of X with the Mordell–Weil rank of the Albanese variety of X K {X_{K}} and the rank of the Néron–Severi group of X K {X_{K}} . This is a higher-dimensional and arithmetic version of the classical Shioda–Tate formula for elliptic surfaces. Such an analogy is strengthened by the fact that we show that the numerically trivial arithmetic ℝ {\mathbb{R}} -divisors on X are exactly the linear combinations of principal ones. This result is equivalent to the non-degeneracy of the arithmetic intersection pairing in the argument of divisors, partially confirming a conjecture by H. Gillet and C. Soulé.

Special cycles on unitary Shimura varieties with minuscule parahoric level structure

In this paper, we formulate conjectural formulas for the arithmetic intersection numbers of special cycles on unitary Shimura varieties with minuscule parahoric level structure. Also, we prove that these conjectures are compatible with all known results.

On arithmetic intersection numbers on self-products of curves

We give a closed formula for the Néron–Tate height of tautological integral cycles on Jacobians of curves over number fields as well as a new lower bound for the arithmetic self-intersection number ω ^ 2 \hat {\omega }^2 of the dualizing sheaf of a curve in terms of Zhang’s invariant φ \varphi . As an application, we obtain an effective Bogomolov-type result for the tautological cycles. We deduce these results from a more general combinatorial computation of arithmetic intersection numbers of adelic line bundles on higher self-products of curves, which are linear combinations of pullbacks of line bundles on the curve and the diagonal bundle.

Kudla–Rapoport cycles and derivatives of local densities

We prove the local Kudla–Rapoport conjecture, which is a precise identity between the arithmetic intersection numbers of special cycles on unitary Rapoport–Zink spaces and the derivatives of local representation densities of hermitian forms. As a first application, we prove the global Kudla–Rapoport conjecture, which relates the arithmetic intersection numbers of special cycles on unitary Shimura varieties and the central derivatives of the Fourier coefficients of incoherent Eisenstein series. Combining previous results of Liu and Garcia–Sankaran, we also prove cases of the arithmetic Siegel–Weil formula in any dimension.

Arithmetic intersections of modular geodesics

The arithmetic, p-arithmetic, and incoherent intersections between pairs of closed geodesics on a modular or Shimura curve are defined, and some of their expected algebraicity and factorisation properties are examined. These properties follow in a special case from the conjectures of [DV] on the RM values of rigid meromorphic cocycles, and are inspired from the recent work of James Rickards [Ri] and Xavier Guitart, Marc Masdeu and Xavier Xarles [GMX] in the quaternionic setting.

The arithmetic Hodge index theorem and rigidity of dynamical systems over function fields

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the N\'eron-Tate height pairing. More recently, Moriwaki and Yuan-Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow's $K/k$-trace functor. As an application of the Hodge Index Theorem, we also prove a rigidity theorem for the set of canonical height zero points of polarized algebraic dynamical systems over function fields. For function fields over finite fields, this gives a rigidity theorem for preperiodic points, generalizing previous work of Mimar, Baker-DeMarco, and Yuan-Zhang.

Arithmetic diagonal cycles on unitary Shimura varieties

We define variants of PEL type of the Shimura varieties that appear in the context of the arithmetic Gan–Gross–Prasad (AGGP) conjecture. We formulate for them a version of the AGGP conjecture. We also construct (global and semi-global) integral models of these Shimura varieties and formulate for them conjectures on arithmetic intersection numbers. We prove some of these conjectures in low dimension.

Riemann–Roch isometries in the non-compact orbifold setting

We generalize work of Deligne and Gillet–Soulé on a functorial Riemann–Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces of finite type \Gamma\backslash\mathbb H , for \Gamma\subset\mathrm {PSL}_{2}(\mathbb R) a Fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cusps and elliptic fixed points, and the original results of Deligne and Gillet–Soulé do not apply to this setting. Our theorem relates the determinant of cohomology of the trivial sheaf, with an explicit Quillen type metric defined in terms of the Selberg zeta function of \Gamma , to a metrized version of the tautological line bundle in the theory of moduli spaces of pointed orbicurves, and the self-intersection bundle of a suitable twist of the canonical sheaf \omega . We make use of surgery techniques through Mayer–Vietoris formulas for determinants of Laplacians, in order to reduce to explicit evaluations of such for model hyperbolic cusps and cones. We carry out these computations, which are of independent interest for theoretical physics, and can be adapted to other geometries. We go on to derive an arithmetic Riemann–Roch formula in the realm of Arakelov geometry, which applies in particular to integral models of modular curves with elliptic fixed points. This vastly extends previous work of the first author, whose deformation-theoretic methods were limited to torsion free Fuchsian groups. As an application, we treat in detail the case of the modular curve X(1) , and we show the interesting arithmetic content of the metrized \Psi line bundles. From this, we answer the longstanding question of evaluating the Selberg zeta special value Z'(1, \mathrm {PSL}_2 (\mathbb Z)) . The result is expressed in terms of logarithmic derivatives of Dirichlet L functions. In the analogy between Selberg zeta functions and Dedekind zeta functions of number fields, this formula can be seen as the analytic class number formula for Z(s, \mathrm {PSL}_2 (\mathbb Z)) . The methods developed in this article were conceived so that they afford several variants, such as the determinant of cohomology of a flat unitary vector bundle with finite monodromies at cusps. Our work finds its place in the program initiated by Burgos–Kramer–Kühn of extending arithmetic intersection theory to singular Hermitian vector bundles.

Modularity of generating series of divisors on unitary Shimura varieties II: arithmetic applications

We prove two formulas in the style of the Gross-Zagier theorem, relating derivatives of L-functions to arithmetic intersection pairings on a unitary Shimura variety. We also prove a special case of Colmez's conjecture on the Faltings heights of abelian varieties with complex multiplication. These results are derived from the authors' earlier results on the modularity of generating series of divisors on unitary Shimura varieties.

Explicit arithmetic intersection theory and computation of Néron-Tate heights

We describe a general algorithm for computing intersection pairings on arithmetic surfaces. We have implemented our algorithm for curves over Q, and we show how to use it to compute regulators for a number of Jacobians of smooth plane quartics, and to numerically verify the conjecture of Birch and Swinnerton-Dyer for the Jacobian of the split Cartan curve of level 13, up to squares.

Higher arithmetic intersection theory

We give a new definition of higher arithmetic Chow groups for smooth projective varieties defined over an arithmetic field, which is similar to Gillet and Soulé's definition of arithmetic Chow groups. We also give a compact description of the intersection theory of such groups. A consequence of this theory is the definition of a heigh pairing between two higher algebraic cycles of complementary dimensions, whose real regulator class is zero. This description agrees with Beilinson's height pairing for the classical arithmetic Chow groups. We also give examples of higher arithmetic intersection pairing in dimension zero.

FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

Flat line bundles and the Cappell-Miller torsion in Arakelov geometry

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology side, the idea goes back to Fay's holomorphic extension of determinants of Dolbeault laplacians, and it is shown here to be equivalent to the holomorphic Cappell-Miller torsion. On the Deligne pairing side, the logarithm is a refinement of the intersection connections considered in previous work. The construction naturally leads to an Arakelov theory for flat line bundles on arithmetic surfaces and produces arithmetic intersection numbers valued in ${\mathbb C}/\pi i {\mathbb Z}$. In this context we prove an arithmetic Riemann-Roch theorem. This realizes a program proposed by Cappell-Miller to show that the holomorphic torsion exhibits properties similar to those of the Quillen metric proved by Bismut, Gillet and Soule. Finally, we give examples that clarify the kind of invariants that the formalism captures; namely, periods of differential forms.

Green forms and the arithmetic Siegel–Weil formula

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.

On two arithmetic theta lifts

Our aim is to clarify the relationship between Kudla’s and Bruinier’s Green functions attached to special cycles on Shimura varieties of orthogonal and unitary type, which play a key role in the arithmetic geometry of these cycles in the context of Kudla’s program. In particular, we show that the generating series obtained by taking the differences of the two families of Green functions is a non-holomorphic modular form and has trivial (cuspidal) holomorphic projection. Along the way, we construct a section of the Maaß lowering operator for moderate growth forms valued in the Weil representation using a regularized theta lift, which may be of independent interest, as it in particular has applications to mock modular forms. We also consider arithmetic-geometric applications to integral models of$U(n,1)$Shimura varieties. Each family of Green functions gives rise to a formal arithmetic theta function, valued in an arithmetic Chow group, that is conjectured to be modular; our main result is the modularity of the difference of the two arithmetic theta functions. Finally, we relate the arithmetic heights of the special cycles to special derivatives of Eisenstein series, as predicted by Kudla’s conjecture, and describe a refinement of a theorem of Bruinier, Howard and Yang on arithmetic intersections against CM points.

An arithmetic Bernštein–Kušnirenko inequality

We present an upper bound for the height of the isolated zeros in the torus of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bernstein–Kusnirenko theorem. Its proof is based on arithmetic intersection theory on toric varieties.