Explicit CM theory for level 2-structures on abelian surfaces

We present a new method for constructing simple ordinary abelian surfaces with a small embedding degree. To a quartic CM field K, we associate a quadric surface H ⊂ ℙ3(ℚ) and use its parametrization to determine Weil numbers in K corresponding in the sense of Honda-Tate theory to such surfaces. In general, the resulting surfaces have parameter ρ ≈ 8. However, if there exist rational lines on H, they can be used to achieve ρ ≈ 4. We give examples of non-primitive quartic CM fields such that H has rulings by rational lines. Furthermore, we show how our method can be used to construct parametric families of pairing-friendly surfaces.

This is a survey article about Siegel modular varieties over the complex numbers. It is written mostly from the point of view of moduli of abelian varieties, especially surfaces. We cover compactification of Siegel modular varieties; classification of the compactified varieties by Kodaira dimension, etc.; moduli of abelian surfaces and especially applications of the lifting of Jacobi forms to modular forms; projective models of some special Siegel modular 3-folds; non-principally polarized abelian surfaces; and constructing degenerating families of abelian varieties.

The Colmez conjecture relates the Faltings height of an abelian variety with complex multiplication by the ring of integers of a CM field E to logarithmic derivatives of Artin L-functions at $$s=0$$ . In this paper, we prove that if F is any fixed totally real number field of degree $$[F:\mathbb {Q}] \ge 3$$ , then there are infinitely many effective, “positive density” sets of CM extensions E / F such that $$E/\mathbb {Q}$$ is non-abelian and the Colmez conjecture is true for E. Moreover, these CM extensions are explicitly constructed to be ramified at arbitrary prescribed sets of prime ideals of F. We also prove that the Colmez conjecture is true for a generic class of non-abelian CM fields called Weyl CM fields, and use this to develop an arithmetic statistics approach to the Colmez conjecture based on counting CM fields of fixed degree and bounded discriminant. We illustrate these results by evaluating the Faltings height of the Jacobian of a genus 2 hyperelliptic curve with complex multiplication by a non-abelian quartic CM field in terms of the Barnes double Gamma function at algebraic arguments. This can be viewed as an explicit non-abelian Chowla–Selberg formula. Our results rely crucially on an averaged version of the Colmez conjecture which was recently proved independently by Andreatta–Goren–Howard–Madapusi Pera and Yuan–Zhang.

We describe the structure of Tate-Shafarevich groups of a constant elliptic curves over function fields by exploiting the volcano structure of isogeny graphs of elliptic curves over finite fields.

For an abelian surface A over a number field k , we study the limiting distribution of the normalized Euler factors of the L -function of A . This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato–Tate group may be obtained from the Galois action on any Tate module of A . We show that the Sato–Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the ℝ-algebra generated by endomorphisms of $A_{{\overline {\mathbb Q}}}$ (the Galois type ), and establish a matching with the classification of Sato–Tate groups; this shows that there are at most 52 groups up to conjugacy which occur as Sato–Tate groups for suitable A and k , of which 34 can occur for k =ℚ. Finally, we present examples of Jacobians of hyperelliptic curves exhibiting each Galois type (over ℚ whenever possible), and observe numerical agreement with the expected Sato–Tate distribution by comparing moment statistics.

Consider a quartic q-Weil polynomial ƒ. Motivated by equidistribution considerations, we define, for each prime ℓ, a local factor that measures the relative frequency with which ƒ mod ℓ occurs as the characteristic polynomial of a symplectic similitude over 𝔽ℓ. For a certain class of polynomials, we show that the resulting infinite product calculates the number of principally polarized abelian surfaces over 𝔽q with Weil polynomial ƒ.

We examine the relationship between the notion of Frobe- nius splitting and ordinarity for varieties. We show the following: a) The de Rham-Witt cohomology groups H i (X, W(OX)) of a smooth projec- tive Frobenius split variety are finitely generated over W(k). b) we provide counterexamples to a question of V. B. Mehta that Frobenius split varieties are ordinary or even Hodge-Witt. c) a Kummer K3 sur- face associated to an Abelian surface is F-split (ordinary) if and only if the associated Abelian surface is F-split (ordinary). d) for a K3-surface defined over a number field, there is a set of primes of density one in some finite extension of the base field, over which the surface acquires ordinary reduction.

In this paper we treat in details a Siegel modular variety [Formula: see text] that has a Calabi–Yau model, [Formula: see text]. We shall describe the structure of the ring of modular forms and its geometry. We shall illustrate two different methods of producing the Hodge numbers. The first uses the definition of [Formula: see text] as the quotient of another known Calabi–Yau variety [Formula: see text]. In this case we will get the Hodge numbers considering the action of the group on a crepant resolution [Formula: see text] of [Formula: see text]. The second, purely algebraic geometric, uses the equations derived from the ring of modular forms and is based on determining explicitly the Calabi–Yau model [Formula: see text] and computing the Picard group and the Euler characteristic.

We present algorithms for computing j j -invariants, modular polynomials and explicit isogenies for ordinary rank 2 Drinfeld modules over finite fields and describe how Drinfeld modular polynomials can be used to compute isogeny graphs and endomorphism rings of ordinary rank 2 Drinfeld modules. Our technique for computing Drinfeld modular polynomials is based on the traditional analytic approach for obtaining classical modular polynomials. Our ideas for generating isogeny graphs and finding endomorphism rings for rank 2 Drinfeld modules closely follows the work of Kohel and Fouquet. All our algorithms were implemented in SAGE and numerical examples are included.

A method for polynomial multiplication over finite fields using field extensions and polynomial interpolation is introduced. The proposed method uses polynomial interpolation as Toom-Cook method together with field extensions. Furthermore, the proposed method can be used when Toom-Cook method cannot be applied directly. Explicit formulae improving the previous results in many cases are obtained.

This paper studies stable sheaves on abelian surfaces of Picard number one. Our main tools are semi-homogeneous sheaves and Fourier–Mukai transforms. We introduce the notion of semi-homogeneous presentation and investigate the behavior of stable sheaves under Fourier–Mukai transforms. As a consequence, an affirmative answer is given to the conjecture proposed by Mukai in the 1980s. This paper also includes an explicit description of the birational correspondence between the moduli spaces of stable sheaves and the Hilbert schemes.

The theory of polynomials over finite fields is important for investigating the algebraic structure of finite fields as well as for many applications. Above all, irreducible polynomials—the prime elements of the polynomial ring over a finite field—are indispensable for constructing finite fields and computing with the elements of a finite field. Section 1 introduces the notion of the order of a polynomial. An important fact is the connection between minimal polynomials of primitive elements (so-called primitive polynomials) and polynomials of the highest possible order for a given degree. Results about irreducible polynomials going beyond those discussed in the previous chapters are presented in Section 2. The next section is devoted to constructive aspects of irreducibility and deals also with the problem of calculating the minimal polynomial of an element in an extension field. Certain special types of polynomials are discussed in the last two sections. Linearized polynomials are singled out by the property that all the exponents occurring in them are powers of the characteristic. The remarkable theory of these polynomials enables us, in particular, to give an alternative proof of the normal basis theorem. Binomials and trinomials — that is, two-term and three-term polynomials—form another class of polynomials for which special results of considerable interest can be established. We remark that another useful collection of polynomials— namely, that of cyclotomic polynomials—was already considered in Chapter 2, Section 4, and that some additional information on cyclotomic polynomials is contained in Section 2 of the present chapter.

We give a one-parameter family of Bridgeland stability conditions on the derived category of a smooth projective complex surface S and describe "wall-crossing behavior'' for objects with the same invariants as \mathcal O_C(H) when H generates Pic (S) and C \in |H| . If, in addition, S is a K3 or Abelian surface, we use this description to construct a sequence of fine moduli spaces of Bridgeland-stable objects via Mukai flops and generalized elementary modifications of the universal coherent sheaf. We also discover a natural generalization of Thaddeus' stable pairs for curves embedded in the moduli spaces.

We give a new algorithm for constructing Picard curves over a finite field with a given endomorphism ring. This has important applications in cryptography since curves of genus 3 allow for smaller key sizes than elliptic curves. For a sextic CM-field $K$ containing the cube roots of unity, we define and compute certain class polynomials modulo small primes and then use the Chinese Remainder Theorem to construct the class polynomials over the rationals. We also give some examples.

Generalizing a method of Sutherland and the author for elliptic cur ves we design a subexponential algorithm for computing the endomorphism rings of ordinary abelian varieties of dimension two over finite fields. Although its correctness and complexity analysis rest on several assumptions, we report on practical computations showing that it performs very well and can easily handle previously intractable cases. Note . Some results of this paper previously appeared in the author’s thesis, [ Endomorphism Rings in Cryptography , Ph.D. Thesis. Eindhoven University of Technology and Institut National Polytechnique de Lorraine, 2011. ISBN: 90-386-2519-7].