JAMI Workshop: Riemann-Roch in Characteristic One and Related Topics

JAMI Workshop: Riemann-Roch in characteristic one and related topics

Titles and Abstracts

Title: Pastures, Polynomials, and Matroids - M. Baker Abstract: A pasture is, roughly speaking, a hyperfield in which addition is allowed to be not only multivalued but also partially undefined. Pastures are natural objects from the point of view of F1 geometry and Lorscheid’s theory of ordered blueprints, and they form a category possessing both products and co-products as well as initial and final objects. I will describe the theory of univariate polynomials over pastures, which simultaneously generalizes both Descartes’ Rule of Signs and the theory of Newton Polygons. I will also describe a novel approach to the theory of matroid representations which revolves around a canonical universal pasture, called the “foundation”, that one can attach to any matroid. This is joint work with Nathan Bowler and Oliver Lorscheid.

Title: The Braid Groups and Descent from the Integers to K(F1) - J. Beardsley Abstract: I will talk about joint work with Jack Morava in which we describe the Eilenberg-MacLane spectrum HZ as a quotient of K(F1) ' S by an action of the braids of trivial writhe. It follows that HZ may be described as a Hopf- Galois extension of K(F1) descent along which is controlled by the braid groups.

Title: The étalefundamental group for semirings (following Culling) - J. Borger Abstract: I’ll explain an extension of the basic theory of the étalefundamental group from rings to semirings, as developed in Robert Culling’s recent ANU PhD thesis. I’ll also explain his proof that the semirings B (Booleans), R+ (non-negative reals), and N (natural numbers) all have trivial étalefundamental groups, much as C and Z do in usual scheme theory. I’ll suggest some open questions along the way.

Title: Riemann-Roch in characteristic 1 and absolute algebraic geometry - A. Connes Abstract: I will explain joint work with C. Consani going from the Riemann- Roch theorem for periodic orbits of the scaling site to the unifying framework of

1 algebraic geometry of Segal’s Gamma rings. This new framework allows for the development of homological algebra thanks to the existing theories of Gamma spaces and of topological cyclic homology.

Title: From tropical to ambitropical convexity - S. Gaubert Abstract: Closed tropical convex cones are the most basic examples of modules over the tropical semifield. They coincide with sub-fixed-point sets of Shapley operators – dynamic programming operators of zero-sum games. We study a larger class of cones, which we call “ambitropical” as it includes both tropical cones and their duals. Ambitropical cones can be defined as lattices in the order induced by Rn. Closed ambitropical cones are precisely the fixed- point sets of Shapley operators. They are characterized by a property of best co-approximation arising from the theory of nonexpansive retracts of normed spaces. Finitely generated ambitropical cones arise when considering Shapley operators of deterministic games with finite action spaces. Finitely generated ambitropical cones are special polyhedral complexes whose cells are alcoved poyhedra, and locally, they are in bijection with order preserving retracts of the Boolean cube. This talk is based on joint work with M. Akian and S. Vannucci.

Title: Directed topology, and strongly non-abelian homology and homotopy theories - E. Goubault Abstract: In this talk, I will develop a homology and homotopy theory for directed spaces, with the hope to foster connections to homological algebra in characteristic one. This will mostly take the form of a survey of recent results in directed topology, a branch of topology which considers topological spaces that have a preferred “flow of time”. In particular, I will develop a homology theory which gives a homological category in the sense of Grandis, providing fine invariants for directed topological spaces in terms of natural systems àla Baues-Wirshing. If time permits, I will develop also some of the homotopical side of the story, refining the structure of directed invariants.

Title: An arithmetic topos for integer matrices and other monoids - J. Hemelaer Abstract: We consider a variation on the Connes–Consani Arithmetic Site ns given by the topos M2 (Z)-Sets of sets with a left action by the monoid

ns M2 (Z) = {a ∈ M2(Z) : det(a) 6= 0}. We show that the points of this topos are classiﬁed by the double quotient

f GL2(Zb) \ M2(A ) / GL2(Q), or alternatively, by the rank two subgroups of Q2 up to isomorphism. We compute the topos automorphisms and relate the combinatorics of the topos to Conway’s big picture and to Goormaghtigh conjecture. Finally, we discuss to what extent the methods can be generalized to other monoids, as some kind of noncommutative algebraic geometry over F1.

2 Title: The Hall algebra of the category of matroids - J. Jun Abstract: To an abelian category A satisfying certain finiteness conditions, one can associate an algebra HA (the Hall algebra of A) which encodes the structures of the space of extensions between objects in A. For a non-additive setting, Dyckerhoff and Kapranov introduced the notion of proto-exact categories, as a non-additive generalization of an exact category, which is shown to suffice for the construction of an associative Hall algebra. In this talk, I will discuss the category of matroids in this perspective.

Title: Convergence of Euler products of the absolute tensor products of L- functions - Shin-ya Koyama Abstract: Let 0 < α < 1/2. For distinct prime numbers p, q and Dirichlet characters χ1, χ2, the (p, q)-Euler factor of the Kurokawa’s absolute tensor product of Dirichlet L-functions L(s, χ1) and L(s, χ2) is given by

m n 1 X χ1(p) χ2(q) (log p)(log q) Lα (s, χ , χ ) = exp p,q 1 2 πi (m log p)2 − (n log q)2 m,n cosh(mα log p) −m(s− 1 ) n log q sinh(mα log p) −m(s− 1 ) p 2 + p 2 n(α+ 1 ) n(α+ 1 ) q 2 m log p q 2 ! m log p sinh(nα log q) −n(s− 1 ) cosh(nα log q) −n(s− 1 ) − q 2 − q 2 . m(α+ 1 ) m(α+ 1 ) n log q p 2 p 2

The (double) Euler product over all distinct pairs of primes

Y α Lp,q(s, χ1, χ2)(∗) p6=q gives the essential part of the absolute tensor product L(s, χ1) ⊗ L(s, χ2). In 2005, Kurokawa and Koyama proved that it is absolutely convergent in <(s) > 2, having zeros at sums of zeros of L(s, χ1) and L(s, χ2). In this talk, we prove that the (double) Euler product (∗) is convergent in <(s) ≥ 1 + 2α under the assumption that the Euler products of L(s, χ1) and L(s, χ2) are convergent in <(s) ≥ (1/2) + α. This assumption is reasonable in the context of the Deep Riemann Hypothesis (DRH) named by Kurokawa.

Title: The directed algebraic topology of monoids - S. Krishnan Abstract: This talk describes a model structure in directed homotopy that makes it possible for doing homological algebra in characteristic one. The idea is to fully embed monoids into a homotopy category coming from two different, but equivalent, model categories of based directed spaces. In one of these model categories, every object is cofibrant and the fibrant objects are models of higher dimensional preordered groups. In the other of these model categories, every object is fibrant and the cofibrant objects include closed smooth manifolds equipped with locally constant, free, and generating cones on their tangent bundles. This talk describes a cohomology theory, taking coefficients and values

3 in semigroups satisfying a weak commutativity condition, on directed spaces, and how to bootstrap such a theory for sheaves of semilattices.

Title: Towards tropical Riemann Roch - hyperfields, blue schemes and matroid bundles - O. Lorscheid Abstract: In this talk, we highlight a few key concepts in an attempt to develop a cohomological understanding of the tropical Riemann Roch theorem. Our story begins with the insight that ordered blue schemes provide a satisfying language for algebraic geometry over the tropical hyperfield. This allows us, for instance, to understand the tropicalization of a classical variety as the base change to the tropical hyperfield. It also justifies the role of the Giansiracusa bend relations for tropical scheme theory. When attempting to develop sheaf cohomology for tropical schemes, matroids as a replacement of linear algebra are inevitable. The geometric coun- terpart are matroid bundles, which appear naturally in the study of the moduli space of matroids in a joint work with Matthew Baker. We will explain why this theory goes hand in hand with algebraic geometry over the tropical hyperfield.

Title: Prime tropical ideals - K. Mincheva Abstract: Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inher- ently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a non-zero prime tropical ideal is either empty or consists of a single point. This is joint work with D. Joó.

Title: Tropical scheme theory - D. Maclagan Abstract: Tropical geometry can be viewed as algebraic geometry over the tropical semiring (R union infinity, with operations min and +). This perspective has proved surprisingly effective over the last decade, but has so far has mostly been restricted to the study of varieties and cycles. I will discuss a pro- gram to construct a scheme theory for tropical geometry. This builds on schemes over semirings, but also introduces concepts from matroid theory. This is joint work with Felipe Rincon, involving also work of Jeff and Noah Giansiracusa and others.

Title: 3-dimensional foliated dynamical systems and reciprocity law - M. Mor- ishita

4 Abstract: Motivated by the analogies between knots and primes, 3-manifolds and number rings in arithmetic topology, we present some fundamental results on 3-dimensional foliated dynamical systems (FDS for short) introduced by C. Deninger. Firstly, we give a decomposition theorem and a complete classiﬁ- cation for FDS’s. Then we construct concrete examples of FDS for each type of the classiﬁcation. Next, by using the integration theory for smooth Deligne cohomology, we introduce dynamical analogues of local symbols and Hilbert reciprocity law for an FDS.This is the joint work with Junhyeong Kim, Takeo Noda and Yuji Terashima

Title: An absolute version of Hasse zeta functions - H. Ochiai Abstract: We introduce an absolute version of Hasse zeta functions for F1- algebra of ﬁnite type. We discuss several properties of such a zeta, f.g., meromor- phic continuations and functional equations. This is a joint work with Nobushige Kurokawa.

Title: Towards arithmetic sites at some places - A. Sagnier Abstract: Adèleclass spaces for number fields are, as A. Connes stressed it in 1996, very important for the spectral interpretation of zeroes of Hecke L- functions (and so in particular for the Riemann zeta function or for the Dedekind zeta function of a number field). The semiringed topos Nc×, (Z ∪ −∞, max, +) called by A. Connes and C. Consani the arithmetic site and introduced by them in 2014 provides a algebro-geometric background for the adèleclass space of Q. Thanks to this algebro-geometric background, one could hope in the long term to transfer and adapt to the context of number fields (and in A. Connes’ and C. Consani’s case Q) ideas coming from Weil’s proof of the analogue of the Riemann hypothesis in the function field case. In my PhD thesis, I introduced for the number field Q(ı) a semiringed topos Zd[ı], (Z[ı]conv, Conv(∪), +) similar to the one introduced by A. Connes and C. Consani and which is linked to the adèleclass space of Q(ı) and consequently linked to the Dedekind zeta function of Q(ı) and a family of Hecke L-functions. However the question of naturality of the choice of Z[ı]conv as structure sheaf remained unanswered. In this lecture, I will show that Z[ı]conv is the solution to an universal problem coming from the hyperaddition on Z[ı]/{±1, ±ı} and so that this choice of structure sheaf was natural after all. This universal problem coming from an hyperaddition will help us to get hints on√ how to define arithmetic sites for other number fields and for example for Q( 2).

Title: An approximate Riemann-Roch theorem for Euclidean lattices - C. Soul´e Abstract: Given an Euclidean lattice L of rank N and its dual L∗, we show that the number of points of L in its unit ball is equal to the same quantity for L∗, up to the multiplication by a constant depending on N. We discuss to

5 which extent this result is an analog for L of the Riemann-Roch theorem for curves. Joint work with H. Gillet.

Title: On the Euler products of the double Dirichlet L functions - H. Tanaka Abstract: In 1992 Kurokawa deﬁned “absolute tensor products.” For zeta functions Zj(s)(s ∈ C, j = 1, 2, ··· r), the Kurokawa’s absolute tensor product Z1(s) ⊗Kur · · · ⊗Kur Zr(s) has a zero or a pole only at s = ρ1 + ··· + ρr (Zj(ρj) = 0 or ∞). In this talk, we analytically construct the Euler product of the double Dirich- ⊗2 let L function L (s, χ) := L(s, χ) ⊗Kur L(s, χ) for <(s) > 2. For the proof, we need to establish the zeta regularized product expression of the Dirichlet L function L(s, χ) and the explicit formula of a theta type P −τχt series e as well as its functional equation, where τχ ∈ C is deﬁned <(τχ)>0 by ρχ = 1/2 + iτχ for a zero ρχ of the Dirichlet L function L(s, χ).