Abelian Surfaces Research Articles

Unlikely Intersections with ExCM curves in A2

The Zilber-Pink conjecture predicts that an algebraic curve in $\mathcal{A}_2$ has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large Galois orbits hypothesis, we prove the finiteness of the intersection with the special curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication. Furthermore, we show that this large Galois orbits hypothesis holds for curves satisfying a condition on their intersection with the boundary of the Baily-Borel compactification of $\mathcal{A}_2$. More generally, we show that a Hodge generic curve in an arbitrary Shimura variety has only finitely many intersection points with the generic points of a Hecke-facteur family, again under a large Galois orbits hypothesis.

A supersingular coincidence

The 15 primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71 are called the supersingular primes: they occur in several contexts in number theory and also, strikingly, they are the primes that divide the order of the Monster. It is also known that the moduli space of (1, p)-polarised abelian surfaces is of general type for these primes. In this note, we explain that apparently coincidental fact by relating it to other number-theoretic occurences of the supersingular primes.

On Superspecial abelian surfaces over finite fields III

In the paper (J Math Soc Jpn 72(1):303–331, 2020), Tse-Chung Yang and the first two current authors computed explicitly the number \(|\mathrm {SSp}_2(\mathbb {F}_q)|\) of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field \(\mathbb {F}_q\) of even degree over the prime field \(\mathbb {F}_p\). There it was assumed that certain commutative \(\mathbb {Z}_p\)-orders satisfy an étale condition that excludes the primes \(p=2, 3, 5\). We treat these remaining primes in the present paper, where the computations are more involved because of the ramification. This completes the calculation of \(|\mathrm {SSp}_2(\mathbb {F}_q)|\) in the even degree case. The odd degree case was previous treated by Tse-Chung Yang and the first two current authors in (Doc Math 21:1607–1643, 2016). To complete the proof of our main theorem, we give a classification of lattices over local quaternion Bass orders, which is a new input to our previous works.

G-invariant Hilbert schemes on Abelian surfaces and enumerative geometry of the orbifold Kummer surface

For an Abelian surface A with a symplectic action by a finite group G, one can define the partition function for G-invariant Hilbert schemes $$\begin{aligned} Z_{A, G}(q) = \sum _{d=0}^{\infty } e(\text {Hilb}^{d}(A)^{G})q^{d}. \end{aligned}$$We prove the reciprocal \(Z_{A,G}^{-1}\) is a modular form of weight \(\frac{1}{2}e(A/G)\) for the congruence subgroup \(\Gamma _{0}(|G|)\) and give explicit expressions in terms of eta products. Refined formulas for the \(\chi _{y}\)-genera of \(\text {Hilb}(A)^{G}\) are also given. For the group generated by the standard involution \(\tau : A \rightarrow A\), our formulas arise from the enumerative geometry of the orbifold Kummer surface \([A/\tau ]\). We prove that a virtual count of curves in the stack is governed by \(\chi _{y}(\text {Hilb}(A)^{\tau })\). Moreover, the coefficients of \(Z_{A, \tau }\) are true (weighted) counts of rational curves, consistent with hyperelliptic counts of Bryan, Oberdieck, Pandharipande, and Yin.

On the existence of abelian surfaces with everywhere good reduction

Let D ≤ 2000 D \le 2000 be a positive discriminant such that F = Q ( D ) F = \mathbf {Q}(\sqrt {D}) has narrow class number one, and A / F A/F an abelian surface of GL 2 \operatorname {GL}_2 -type with everywhere good reduction. Assuming that A A is modular, we show that A A is either a Q \mathbf {Q} -surface or is a base change from Q \mathbf {Q} of an abelian surface B B such that End Q ⁡ ( B ) = Z \operatorname {End}_\mathbf {Q}(B) = \mathbf {Z} , except for D = 353 , 421 , 1321 , 1597 D = 353, 421, 1321, 1597 and 1997 1997 . In the latter case, we show that there are indeed abelian surfaces with everywhere good reduction over F F for D = 353 , 421 D = 353, 421 and 1597 1597 , which are non-isogenous to their Galois conjugates. These are the first known such examples.

Abelian surfaces over totally real fields are potentially modular

We show that abelian surfaces (and consequently curves of genus 2) over totally real fields are potentially modular. As a consequence, we obtain the expected meromorphic continuation and functional equations of their Hasse–Weil zeta functions. We furthermore show the modularity of infinitely many abelian surfaces A over {mathbf {Q}} with operatorname{End}_{ {mathbf {C}}}A={mathbf {Z}}. We also deduce modularity and potential modularity results for genus one curves over (not necessarily CM) quadratic extensions of totally real fields.

The Severi problem for abelian surfaces in the primitive case

Severi varieties are roughly moduli spaces of curves of a fixed homology class and geometric genus on a projective surface. In this paper, we determine the irreducible components of the Severi varieties of primitive (indivisible) homology class curves on a general abelian surface, answering a question of Knutsen, Lelli-Chiesa, and Mongardi. More specifically, we prove that the irreducible components are completely determined by the maximal factorization of the morphisms from the normalized curves into the abelian surface.

Hitchin fibrations, abelian surfaces, and the P=W conjecture

We study the topology of Hitchin fibrations via abelian surfaces. We establish the P=W conjecture for genus 2 2 curves and arbitrary rank. In higher genus and arbitrary rank, we prove that P=W holds for the subalgebra of cohomology generated by even tautological classes. Furthermore, we show that all tautological generators lie in the correct pieces of the perverse filtration as predicted by the P=W conjecture. In combination with recent work of Mellit, this reduces the full conjecture to the multiplicativity of the perverse filtration. Our main technique is to study the Hitchin fibration as a degeneration of the Hilbert–Chow morphism associated with the moduli space of certain torsion sheaves on an abelian surface, where the symmetries induced by Markman’s monodromy operators play a crucial role.

Seshadri constants on principally polarized abelian surfaces with real multiplication

Seshadri constants on abelian surfaces are fully understood in the case of Picard number one. Little is known so far for simple abelian surfaces of higher Picard number. In this paper we investigate principally polarized abelian surfaces with real multiplication. They are of Picard number two and might be considered the next natural case to be studied. The challenge is to not only determine the Seshadri constants of individual line bundles, but to understand the whole Seshadri function on these surfaces. Our results show on the one hand that this function is surprisingly complex: on surfaces with real multiplication in mathbb {Z}[sqrt{e}] it consists of linear segments that are never adjacent to each other—it behaves like the Cantor function. On the other hand, we prove that the Seshadri function is invariant under an infinite group of automorphisms, which shows that it does have interesting regular behavior globally.

A special configuration of 12 conics and generalized Kummer surfaces

A generalized Kummer surface X obtained as the quotient of an abelian surface by a symplectic automorphism of order 3 contains a $$9\mathbf{A}_{2}$$ -configuration of $$(-2)$$ -curves. Such a configuration plays the role of the $$16\mathbf{A}_{1}$$ -configurations for usual Kummer surfaces. In this paper we construct 9 other such $$9\mathbf{A}_{2}$$ -configurations on the generalized Kummer surface associated to the double cover of the plane branched over the sextic dual curve of a cubic curve. The new $$9\mathbf{A}_{2}$$ -configurations are obtained by taking the pullback of a certain configuration of 12 conics which are in special position with respect to the branch curve, plus some singular quartic curves. We then construct some automorphisms of the K3 surface sending one configuration to another. We also give various models of X and of the generic fiber of its natural elliptic pencil.

Examples of abelian surfaces failing the local–global principle for isogenies

We provide examples of abelian surfaces over number fields K whose reductions at almost all good primes possess an isogeny of prime degree $$\ell $$ ℓ rational over the residue field, but which themselves do not admit a K-rational $$\ell $$ ℓ -isogeny. This builds on work of Cullinan and Sutherland. When $$K=\mathbb {Q}$$ K = Q , we identify certain weight-2 newforms f with quadratic Fourier coefficients whose associated modular abelian surfaces $$A_f$$ A f exhibit such a failure of a local–global principle for isogenies.

Quantitative Reduction Theory and Unlikely Intersections

AbstractWe prove quantitative versions of Borel and Harish-Chandra’s theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila–Zannier strategy to the Zilber–Pink conjecture for the moduli space of principally polarised abelian surfaces. Building on our previous paper, we prove this conjecture under a Galois orbits hypothesis. Finally, we establish the Galois orbits hypothesis for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions.

Nef Cone of a Generalized Kummer 4-fold

In this note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $\mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $\mathrm{rk} \mathrm{NS}(A) = 1$.

On the computation of the endomorphism rings of abelian surfaces

We extend the result of Bisson [4] to compute the endomorphism rings of principally polarized, absolutely simple, and ordinary abelian surfaces defined over finite fields in subexponential time in the size of the base field. The abelian surfaces covered here were excluded in [4]. This is accomplished by using techniques introduced in [17] to efficiently determine overorders for Bass and Gorenstein orders. In addition we show that the endomorphism rings of certain principally polarized, absolutely simple, and ordinary abelian surfaces are not computable in subexponential time with current methods [4,21], including ours.

Unit groups of maximal orders in totally definite quaternion algebras over real quadratic fields

We study a form of refined class number formula (resp. type number formula) for maximal orders in totally definite quaternion algebras over real quadratic fields, by taking into consideration the automorphism groups of right ideal classes (resp. unit groups of maximal orders). For each finite noncyclic group $G$, we give an explicit formula for the number of conjugacy classes of maximal orders whose unit groups modulo center are isomorphic to $G$, and write down a representative for each conjugacy class. This leads to a complete recipe (even explicit formulas in special cases) for the refined class number formula for all finite groups. As an application, we prove the existence of superspecial abelian surfaces whose endomorphism algebras coincide with $\mathbb{Q}(\sqrt{p})$ in all positive characteristic $p\not\equiv 1\pmod{24}$.

Supersingular O’Grady Varieties of Dimension Six

Abstract O’Grady constructed a 6-dimensional irreducible holomorphic symplectic variety by taking a crepant resolution of some moduli space of stable sheaves on an abelian surface. In this paper, we naturally extend O’Grady’s construction to fields of positive characteristic $p\neq 2$, called OG6 varieties. Assuming $p\geq 3$, we show that a supersingular OG6 variety is unirational, its rational cohomology group is generated by algebraic classes, and its rational Chow motive is of Tate type. These results confirm in this case the generalized Artin–Shioda conjecture, the supersingular Tate conjecture and the supersingular Bloch conjecture proposed in our previous work, in analogy with the theory of supersingular K3 surfaces.

ACM bundles on a general abelian surface

In this note, we shall study semistable aCM bundles on an abelian surface of Picard number 1.

On a classification of the automorphism groups of polarized abelian surfaces over finite fields

We give a classification of maximal elements of the set of finite groups that can be realized as the full automorphism groups of polarized abelian surfaces over finite fields.

On superspecial abelian surfaces and type numbers of totally definite quaternion algebras

In this paper we determine the number of endomorphism rings of superspecial abelian surfaces over a field $\mathbb{F}_q$ of odd degree over $\mathbb{F}_p$ in the isogeny class corresponding to the Weil $q$-number $\pm\sqrt{q}$. This extends earlier works of T.-C. Yang and the present authors on the isomorphism classes of these abelian surfaces, and also generalizes the classical formula of Deuring for the number of endomorphism rings of supersingular elliptic curves. Our method is to explore the relationship between the type and class numbers of the quaternion orders concerned. We study the Picard group action of the center of an arbitrary $\mathbb{Z}$-order in a totally definite quaternion algebra on the ideal class set of said order, and derive an orbit number formula for this action. This allows us to prove an integrality assertion of Vign\'eras [Enseign. Math. (2), 1975] as follows. Let $F$ be a totally real field of even degree over $\mathbb{Q}$, and $D$ be the (unique up to isomorphism) totally definite quaternion $F$-algebra unramified at all finite places of $F$. Then the quotient $h(D)/h(F)$ of the class numbers is an integer.

On uniformity conjectures for abelian varieties and K3 surfaces

We discuss logical links among uniformity conjectures concerning K3 surfaces and abelian varieties of bounded dimension defined over number fields of bounded degree. The conjectures concern the endomorphism algebra of an abelian variety, the Neron-Severi lattice of a K3 surface, and the Galois invariant subgroup of the geometric Brauer group.