Picard Curves Research Articles

Geometry from integrability: multi-leg fishnet integrals in two dimensions

We generalise the geometric analysis of square fishnet integrals in two dimensions to the case of hexagonal fishnets with three-point vertices. Our results support the conjecture that fishnet Feynman integrals in two dimensions, together with their associated geometry, are completely fixed by their Yangian and permutation symmetries. As a new feature for the hexagonal fishnets, the star-triangle identity introduces an ambiguity in the graph representation of a given Feynman integral. This translates into a map between different geometric interpretations attached to a graph. We demonstrate explicitly how these fishnet integrals can be understood as Calabi-Yau varieties, whose Picard-Fuchs ideals are generated by the Yangian over the conformal algebra. In analogy to elliptic curves, which represent the simplest examples of fishnet integrals with four-point vertices, we find that the simplest examples of three-point fishnets correspond to Picard curves with natural generalisations at higher loop orders.

The rank of the Cartier operator on Picard curves

For an algebraic curve [Formula: see text] defined over an algebraically closed field of characteristic [Formula: see text], [Formula: see text]-number [Formula: see text] is the dimension of the space of exact holomorphic differentials on [Formula: see text]. We computed the [Formula: see text]-number for a family of certain Picard curves using the action of the Cartier operator on [Formula: see text].

Tropical invariants for binary quintics and reduction types of Picard curves

Abstract In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$ .

Reduction type of smooth plane quartics

Let $C/K$ be a smooth plane quartic over a discrete valuation field. We characterize the type of reduction (i.e. smooth plane quartic, hyperelliptic genus 3 curve or bad) over $K$ in terms of the existence of a special plane quartic model and, over $\bar{K}$, in terms of the valuations of certain algebraic invariants of $C$ when the characteristic of the residue field is not $2,\,3,\,5$ or $7$. On the way, we gather several results of general interest on geometric invariant theory over an arbitrary ring $R$ in the spirit of (Seshadri 1977). For instance when $R$ is a discrete valuation ring, we show the existence of a homogeneous system of parameters over $R$. We exhibit explicit ones for ternary quartic forms under the action of $\textrm{SL}_{3,R}$ depending only on the characteristic $p$ of the residue field. We illustrate our results with the case of Picard curves for which we give simple criteria for the type of reduction.

Computing 𝐿-polynomials of Picard curves from Cartier–Manin matrices

We study the sequence of zeta functions Z ( C p , T ) Z(C_p,T) of a generic Picard curve C : y 3 = f ( x ) C:y^3=f(x) defined over Q \mathbb {Q} at primes p p of good reduction for C C . We define a degree 9 polynomial ψ f ∈ Q [ x ] \psi _f\in \mathbb {Q}[x] such that the splitting field of ψ f ( x 3 / 2 ) \psi _f(x^3/2) is the 2 2 -torsion field of the Jacobian of C C . We prove that, for all but a density zero subset of primes, the zeta function Z ( C p , T ) Z(C_p,T) is uniquely determined by the Cartier–Manin matrix A p A_p of C C modulo p p and the splitting behavior modulo p p of f f and ψ f \psi _f ; we also show that for primes ≡ 1 ( mod 3 ) \equiv 1 \pmod {3} the matrix A p A_p suffices and that for primes ≡ 2 ( mod 3 ) \equiv 2 \pmod {3} the genericity assumption on C C is unnecessary. An element of the proof, which may be of independent interest, is the determination of the density of the set of primes of ordinary reduction for a generic Picard curve. By combining this with recent work of Sutherland, we obtain a practical deterministic algorithm that computes Z ( C p , T ) Z(C_p,T) for almost all primes p ≤ N p \le N using N log ⁡ ( N ) 3 + o ( 1 ) N\log (N)^{3+o(1)} bit operations. This is the first practical result of this type for curves of genus greater than 2.

An inverse Jacobian algorithm for Picard curves

We study the inverse Jacobian problem for the case of Picard curves over {mathbb {C}}. More precisely, we elaborate on an algorithm that, given a small period matrix varOmega in {mathbb {C}}^{3times 3} corresponding to a principally polarized abelian threefold equipped with an automorphism of order 3, returns a Legendre–Rosenhain equation for a Picard curve with Jacobian isomorphic to the given abelian variety. Our method corrects a formula obtained by Koike–Weng (Math Comput 74(249):499–518, 2005) which is based on a theorem of Siegel. As a result, we apply the algorithm to obtain equations of all the isomorphism classes of Picard curves with maximal complex multiplication by the maximal order of the sextic CM-fields with class number at most 4. In particular, we obtain the complete list of maximal CM Picard curves defined over {mathbb {Q}}. In the appendix, Vincent gives a correction to the generalization of Takase’s formula for the inverse Jacobian problem for hyperelliptic curves given in [Balakrishnan–Ionica–Lauter–Vincent, LMS J. Comput. Math., 19(suppl. A):283-300, 2016].

Conductor and discriminant of Picard curves

We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $\mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine the smallest possible conductor a special Picard curve may have. We also collect a database of Picard curves over $\mathbb{Q}$ of small conductor.

A bound on the primes of bad reduction for CM curves of genus 3

We give bounds on the primes of geometric bad reduction for curves of genus 3 3 of primitive complex multiplication (CM) type in terms of the CM orders. In the case of elliptic curves, there are no primes of geometric bad reduction because CM elliptic curves are CM abelian varieties, which have potential good reduction everywhere. However, for genus at least 2 2 , the curve can have bad reduction at a prime although the Jacobian has good reduction. Goren and Lauter gave the first bound in the case of genus 2 2 . In the cases of hyperelliptic and Picard curves, our results imply bounds on primes appearing in the denominators of invariants and class polynomials, which are important for algorithmic construction of curves with given characteristic polynomials over finite fields.

Primes Dividing Invariants of CM Picard Curves

AbstractWe give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.

Constructing Picard curves with complex multiplication using the Chinese remainder theorem

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.

A cubic transformation formula for Appell–Lauricella hypergeometric functions over finite fields

We define a finite field version of Appell–Lauricella hypergeometric functions built from period functions in several variables, paralleling recent development by Fuselier, Long, Ramakrishna, and the last two authors in the single variable case. We develop geometric connections between these functions and the family of generalized Picard curves. In our main result, we use finite field Appell–Lauricella functions to establish a finite field analogue of Koike and Shiga’s cubic transformation for the Appell hypergeometric function $$F_1$$ , proving a conjecture of Long. We use our multivariable period functions to construct formulas for the number of $$\mathbb {F}_p$$ -points on the generalized Picard curves. We also give some transformation and reduction formulas for the period functions, and consequently for the finite field Appell–Lauricella functions.

Picard curves over with good reduction away from 3

Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over $\mathbb{Q}$ with good reduction away from 3, up to $\mathbb{Q}$-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.

A note on some Picard curves over finite fields

In this note we provide a complete classification for the Newton polygon of the Picard curvesy3=x4−xandy3=x4−1 defined over a finite field of characteristic p>3. In fact, we complete the results obtained in [19]. Using these results, we determine all finite fields over which these curves attain the Hasse–Weil upper bound.

A new interpretation of the Shimura curve with discriminant 6 in terms of Picard modular forms

In this paper we give a realization of the Shimura curve for the quaternion algebra over Q with discriminant 6 as a quotient space of the complex upper half plane by the triangle group Δ(3, 6, 6). It is given by the Schwarz map for the Gauss hypergeometric differential equation $${E\left(\frac{1}{6},\frac{1}{3},\frac{2}{3}\right)}$$ . The corresponding abelian surfaces are obtained as isogenous components of the Jacobi varieties of the Picard curves $${C(s): w^3=z(z-1)\break \left(z-\frac{1}{2} (1-s)\right)\left(z-\frac{1}{2} (1+s)\right)}$$ .

A variant of Jacobi type formula for Picard curves

The classical Jacobi formula for the elliptic integrals (Gesammelte Werke I, p. 235) shows a relation between Jacobi theta constants and periods of ellptic curves E ( λ ) : w 2 = z ( z - 1 ) ( z - λ ) . In other words, this formula says that the modular form ϑ 00 4 ( τ ) with respect to the principal congruence subgroup Γ ( 2 ) of PSL ( 2 , Z ) has an expression by the Gauss hypergeometric function F ( 1 / 2 , 1 / 2 , 1 ; 1 - λ ) via the inverse of the period map for the family of elliptic curves E ( λ ) (see Theorem 1.1). In this article we show a variant of this formula for the family of Picard curves C ( λ 1 , λ 2 ) : w 3 = z ( z - 1 ) ( z - λ 1 ) ( z - λ 2 ) , those are of genus three with two complex parameters. Our result is a two dimensional analogy of this context. The inverse of the period map for C ( λ 1 , λ 2 ) is established in [S] and our modular form ϑ 0 3 ( u , v ) (for the definition, see (2.7)) is defined on a two dimensional complex ball D = { 2 Re v + | u | 2 < 0 } , that can be realized as a Shimura variety in the Siegel upper half space of degree 3 by a modular embedding. Our main theorem says that our theta constant is expressed in terms of the Appell hypergeometric function F 1 ( 1 / 3 , 1 / 3 , 1 / 3 , 1 ; 1 - λ 1 , 1 - λ 2 ) .

ISOMORPHISM CLASSES OF GENUS-3 POINTED TRIGONAL CURVES OVER FINITE FIELDS OF CHARACTERISTIC 2

We find all distinct representatives of isomorphism classes of genus-3 pointed trigonal curves and compute the number of isomorphism classes of a special class of genus-3 pointed trigonal curves including that of Picard curves over a finite field F of characteristic 2.

Galois representations attached to Picard curves

Let J ( C ) be the Jacobian of a Picard curve C defined over a number field K containing Q ( ζ 3 ) . We consider the family ( ρ ˜ l ) l of l-adic representations defined by the natural action of the Galois group Gal ( K ¯ / K ) on the l-power torsion of J ( C ) . We show that for a Picard curve C with endomorphism ring Z [ ζ 3 ] the images of these representations are full for all but finitely many primes l. We consider the reduction modulo l of the image of ρ ˜ l , that is, the action of Gal ( K ¯ / K ) on the l-torsion of the Jacobian. This gives a representation ρ l into either GL 3 ( F l ) or into a unitary group over F l 2 , depending on the splitting behavior of l in Q ( ζ 3 ) . It is sufficient to show that the image of ρ l is full in order to show that the image of ρ ˜ l is full.

Explicit bounds of polynomial coefficients and counting points on Picard curves over finite fields

In this paper, we describe an algorithm for computing the order of the Jacobian varieties of Picard curves over finite fields. This is an extension of the algorithm of Matsuo, Chao and Tsujii (MCT) [K. Matsuo, J. Chao, S. Tsujii, An improved baby step algorithm for point counting of hyperelliptic curves over finite fields, in: LNCS vol. 2369, Springer-Verlag, 2005, pp. 461–474] for hyperelliptic curves. We study the characteristic polynomials and the Jacobian varieties of algebraic curves of genus three over finite fields. Based on this, we investigate the explicit computable bounds for coefficients of the characteristic polynomial and improve a part of the baby-step giant-step of the counting points algorithm. Usefulness of the proposed method is illustrated and verified by the simple examples.

Some remarks on the Picard curves over a finite field

Some remarks on the Picard curves over a finite field

Isogeny formulas for the Picard modular form and a three terms arithmetic geometric mean

In this paper we study the Picard modular forms and show a new three terms arithmetic geometric mean (AGM) system. This AGM system is expressed via the Appell hypergeometric function of two variables. The Picard modular forms are expressed via the theta constants, and they give the modular function for the family of Picard curves. Our theta constants are “Neben type” modular forms of weight 1 defined on the complex 2-dimensional hyperball with respect to an index finite subgroup of the Picard modular group. We define a simultaneous 3-isogeny for the family of Jacobian varieties of Picard curves. Our main theorem shows the explicit relations between two systems of theta constants which are corresponding to isogenous Jacobian varieties. This relation induces a new three terms AGM which is a generalization of Borweins' cubic AGM.