The Formalism of the Period Conjecture

We classify the sum‐free subsets of whose density exceeds . This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum‐free subset is maximal with respect to inclusion and aperiodic (in the sense that there is no non‐zero vector satisfying ), then —a bound known to be optimal if , while for there are no such sets.

Motivic periods and Grothendieck arithmetic invariants

Following Kontsevich (see Kontsevich in Operads and motives in deformation quantization. Lett. Math. Phys. 48(1):35–72, 1999), we now introduce another algebra \(\tilde{\mathbb {P}}(k)\) of formal periods from the same data we have used in order to define the actual period algebra of a field in Chap. 11. The main aim of this chapter is to give conceptual interpretation of this algebra of formal periods. We then use it to formulate and discuss the period conjecture.

ABSTRACTWe describe two algorithms that allow to investigate the graph associated with the nilpotent associative -algebras of coclass r for a finite field and a non-negative r. Based on experimental evidence obtained via these algorithms, we conjecture that is virtually periodic for each finite field and each r. If this periodicity conjecture holds, then it suggests that for each finite field and each r the infinitely many nilpotent associative -algebras of coclass r can be classified by a finite set of data.

In earlier work, the second named author described how to extract Darmon-style L-invariants from modular forms on Shimura curves that are special at p. In this paper, we show that these L-invariants are preserved by the Jacquet‐Langlands correspondence. As a consequence, we prove the second named author’s period conjecture in the case where the base field is Q. As a further application of our methods, we use integrals of Hida families to describe Stark‐Heegner points in terms of a certain Abel‐Jacobi map.

We describe higher dimensional generalizations of Ramanujan’s classical differential relations satisfied by the Eisenstein series E 2 E_2 , E 4 E_4 , E 6 E_6 . Such “higher Ramanujan equations” are given geometrically in terms of vector fields living on certain moduli stacks classifying abelian schemes equipped with suitable frames of their first de Rham cohomology. These vector fields are canonically constructed by means of the Gauss-Manin connection and the Kodaira-Spencer isomorphism. Using Mumford’s theory of degenerating families of abelian varieties, we construct remarkable solutions of these differential equations generalizing ( E 2 , E 4 , E 6 ) (E_2,E_4,E_6) , which are also shown to be defined over Z \mathbf {Z} . This geometric framework taking account of integrality issues is mainly motivated by questions in Transcendental Number Theory regarding an extension of Nesterenko’s celebrated theorem on the algebraic independence of values of Eisenstein series. In this direction, we discuss the precise relation between periods of abelian varieties and the values of the above referred solutions of the higher Ramanujan equations, thereby linking the study of such differential equations to Grothendieck’s Period Conjecture. Working in the complex analytic category, we prove “functional” transcendence results, such as the Zariski-density of every leaf of the holomorphic foliation induced by the higher Ramanujan equations.

Évariste Galois' last letter, addressed to Auguste Chevalier, on the eve of the (so-called) duel on May 30, 1832 (which, perhaps, simpler and more accurately described by Alfred, who did not allow a priest to deprive him from the final moments on the following day with his elder brother Évariste, as murder), was written on seven pages and was divided into three memoirs. The first memoir consumes a little less than two pages. It gave rise to what has come to be known as Galois theory (as, in particular, told by Melvin Kiernan). Yet Galois went on with stunningly amazing constructions in the second memoir, which consumed a bit more than two pages. The third (and longest!) memoir begins on the fifth page and remains mysteriously unresolved, yet it undoubtedly inspired Alexander Grothendieck to formulate his period conjecture. The letter is concluded with a paragraph on the latest ``principal contemplations'', concerning ``the applications of the theory of ambiguity to transcendental analysis'', where Galois delivers his last puzzle to us, saying that ``one recognizes immediately lots of expressions to look for''. Unfortunately, the severity of the time pressure upon him permitted only succinct last instructions with no more last examples. Still and disgracefully, many ``historians'' keep on incessantly and mundanely telling us (and each other) that we ought not ``overestimate'' the significance of the letter, which was (contrary to their advice) eloquently and veraciously described by Hermann Weyl as ``the most substantial piece of writing in the whole literature of mankind''!

Complex multiplication symmetry of black hole attractors

Let A / Q be a Jacobian variety and let F be a totally real, tamely ramified, abelian number field. Given a character ψ of F / Q , Deligne’s Period Conjecture asserts the algebraicity of the suitably normalized value L ( A , ψ , 1 ) at s = 1 of the Hasse-Weil-Artin L-function of the ψ -twist of A. We formulate a conjecture regarding the integrality properties of the family of normalized L-values ( L ( A , ψ , 1 ) ) ψ , and its relation to the Tate-Shafarevich group of A over F. We numerically investigate our conjecture through p-adic congruence relations between these values.

The exceptional zero conjecture relates the first derivative of the $p$-adic $L$-function of a rational elliptic curve with split multiplicative reduction at $p$ to its complex $L$-function. Teitelbaum formulated an analogue of Mazur and Tate's refined (multiplicative) version of this conjecture for elliptic curves over the rational function field $\FQ(T)$ with split multiplicative reduction at two places $\fp$ and $\infty$, avoiding the construction of a $\fp$-adic $L$-function. This article proves Teitelbaum's conjecture up to roots of unity by developing Darmon's theory of double integrals over arbitrary function fields. A function field version of Darmon's period conjecture is also obtained.

Let$\mathsf {C}$be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to$\mathsf {C}$. We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras.We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of$\Delta \square \mathrm {A}_r$.When$\mathsf {C}$is of type$\mathrm {A}$, the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their$\mathbb {F}_q$-points we obtain rational functions that are Legendrian link invariants.

Representability of G-functions as rational functions in hypergeometric series

Third kind elliptic integrals and 1-motives

Motivated by the study of algebraic classes in mixed characteristic, we define a countable subalgebra of ${\overline {\mathbb {Q}}}_p$ which we call the algebra of André’s p -adic periods. The classical Tannakian formalism cannot be used to study these new periods. Instead, inspired by ideas of Drinfel’d on the Plücker embedding and further developed by Haines, we produce an adapted Tannakian setting which allows us to bound the transcendence degree of André’s p -adic periods and to formulate the p -adic analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical p -adic functions appear as André’s p -adic periods, and we relate these new conjectures to some classical problems on algebraic classes.

We prove that the existence of an automorphism of finite order on a Q-variety X implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Γ-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne [11, p. 205] 1 . Our proof relies on the arithmetic fixed-point