In the mathematical fields of

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

and

arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties. ...

, the Consani–Scholten quintic is an algebraic

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...

(the set of solutions to a single

polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field (mathematics), field, often the field of the rational numbers. For example, x^5-3x+1=0 is a ...

in multiple variables) studied in 2001 by

Caterina Consani Caterina (Katia) Consani (born 1963) is an Italian mathematician specializing in arithmetic geometry. She is a professor of mathematics at Johns Hopkins University. Contributions Consani is the namesake of the Consani–Scholten quintic, a quin ...

and Jasper Scholten. It has been used as a test case for the

Langlands program In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number t ...

Definition

Consani and Scholten define their hypersurface from the ( projectivized) set of solutions to the equation :

P(x,y)=P(z,w)

in four complex variables, where :

P(x,y)=x^5+y^5-(5xy-5)(x^2+y^2-x-y).

In this form the resulting hypersurface is

singular Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular or sounder, a group of boar, see List of animal names * Singular (band), a Thai jazz pop duo *'' Singula ...

: it has 120

double point In geometry, a singular point on a curve is one where the curve is not given by a smooth embedding of a parameter. The precise definition of a singular point depends on the type of curve being studied. Algebraic curves in the plane Algebraic cur ...

s. Its

Hodge diamond Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory. History In an address ...

is The Consani–Scholton quintic itself is the non-singular hypersurface obtained by

blowing up In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with the space of all directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the poin ...

these singularities. As a non-singular

quintic threefold In mathematics, a quintic threefold is a 3-dimensional hypersurface of degree 5 in 4-dimensional projective space \mathbb^4. Non-singular quintic threefolds are Calabi–Yau manifolds. The Hodge diamond of a non-singular quintic 3-fold is Physi ...

, it is a

Calabi–Yau manifold In algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties, such as Ricci flatness, yielding applications in theoretical physics. P ...

Modularity

According to the Langlands program, for any Calabi–Yau threefold

X

over

\mathbb

, the

Galois representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...

s giving the action of the

absolute Galois group In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' ...

on the

\ell

-adic

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...

H^3(X_,\mathbb_)

(for

prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...

\ell

good reduction In most contexts, the concept of good denotes the conduct that should be preferred when posed with a choice between possible actions. Good is generally considered to be the opposite of evil. The specific meaning and etymology of the term and its ...

, which for this curve means any prime other than 2, 3, or 5) should have the same L-series as an

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G o ...

. This was known for "rigid" Calabi–Yau threefolds, for which the family of Galois representations has dimension two, by the proof of

Serre's modularity conjecture In mathematics, Serre's modularity conjecture, introduced by , states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and ...

. The Consani–Scholton quintic provides a non-rigid example, where the dimension is four. Consani and Scholten constructed a

Hilbert modular form In mathematics, a Hilbert modular form is a generalization of modular forms to functions of two or more variables. It is a (complex) analytic function on the ''m''-fold product of upper half-planes \mathcal satisfying a certain kind of functional ...

and conjectured that its L-series agreed with the Galois representations for their curve; this was proven by .

References

{{DEFAULTSORT:Consani-Scholten quintic 3-folds