Thinking Positive: in 푝 Renee Bell, Julia Hartmann, Valentijn Karemaker, Padmavathi Srinivasan, and Isabel Vogt

of prime characteristic 푝, like finite fields. Decades after The authors of this piece are organizers of the AMS these ideas were formalized, characteristic 푝 arithmetic ge- 2019 Research Communities summer ometry is rapidly expanding to include work in the vibrant conference on Explicit Methods in Arithmetic young fields of and derived in Characteristic p, one of three topical geometry. research conferences offered this year that are focused Fields of positive characteristic 푝 have a fundamentally on collaborative research and professional distinct flavor from the classical setting of fields of charac- development for early career mathematicians. teristic 0. In , having to let go of the archimedean framework requires a radically different geometric intuition. Additional information can be found at 푝 http://www.ams.org/programs/research- Working in characteristic therefore comes with additional challenges, but there are also additional tools and struc- communities/2019MRC-Explicit. ture that can be exploited, which have led to remarkable Applications are open until February 15, 2019. results. In some instances these results even carry over to 0 arose as a beautiful and powerful the- solve open problems in characteristic . ory unifying geometry and , formalizing striking analogies between them in a way that allowed Historical overview. While number theorists had been tools, results, and intuition of each to be transported to studying the rational and other number fields us- the other—a notable example of this is the proof of the ing the Riemann and Dedekind zeta functions, E. Artin in centuries-old problem, Fermat’s Last Theorem. This the- his 1921 thesis [1] first proposed an analogous theory of ory provides a geometric viewpoint of objects over fields zeta functions for over finite fields. Hasse extended this to prove the for elliptic curves Renee Bell is a Hans Rademacher Instructor at the of Pennsylvania. over finite fields. Their work further sparked the develop- Her email address is [email protected] ment of the theory of fields, as well as the theory Julia Hartmann is a professor at the University of Pennsylvania. Her email of algebraic varieties, by several mathematicians, includ- address is [email protected] ing Weil. In the 1940’s, Weil published a book [11] on Valentijn Karemaker is a postdoctoral research fellow at the University of Penn- the foundations of algebraic geometry in characteristic 푝, sylvania. Her email address is [email protected] and subsequently proved the Riemann hypothesis for func- Padmavathi Srinivasan is a Hale Visiting Assistant Professor at the Georgia tion fields over finite fields [12]. Moreover, in 1949, Weil Institute of Technology. Her email address is padmavathi.srinivasan @math.gatech.edu. proposed conjectures on the behavior of zeta functions of Isabel Vogt is a Massachusetts Institute of Technology graduate student currently varieties over finite fields—including the Riemann visiting Stanford University. Her email address is [email protected] hypothesis—which came to be known as the Weil conjec- For permission to reprint this article, please contact: tures [13]. The breakthrough which enabled Grothendieck, [email protected] DOI: https://doi.org/10.1090/noti1804

FEBRUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 239 M. Artin, and Verdier to prove all but one of the Weil con- behavior of ranks and other fundamental invariants of el- jectures in 1964 [5] was their development of ℓ-adic coho- liptic curves. mology. Building on this, the Riemann hypothesis was eventually proven in two different ways by Deligne ([3, 1974], [4, 1980]). For more historical details, see e.g. [8]. Example 2: Automorphisms. A broad theme in mathemat- ics is that one should study the (i.e. automor- and points. The zeta function phisms) of an object alongside the object itself. In the sim- 푍푋(푇) of an 푋 over 픽푞 encompasses in- plest case of an algebraic over ℂ, the automorphism formation about counts over all finite extensions of varies by the topological type: it is infinite for curves 픽푞: of 푔 = 0 or 1, but once 푔 ≥ 2, Hurwitz famously 84(푔 − 1) 푇푛 proved that its order is at most . In character- 푍푋(푇) = exp ( ∑ #푋(픽푞푛 ) ) . istic 푝, an could have additional symme- 푛 푛≥0 tries! For example, the projective curve 푥푝+1 +푦푧푝 + 푝 8 The Weil conjectures predict that the power 푍푋(푇) 푧푦 over 픽푝2 has genus 푝(푝 − 1)/2, but 푂(푝 ) symme- is a , whose zeros and poles can be de- tries. These symmetries arise in analogy with the fact that scribed in terms of natural group actions on the associated the unitary group PGU3(ℂ) preserves the Hermitian form 푋 = ℙ1 푄(푥, 푦, 푧) = 푥 ̄푥+ 푦 ̄푧+ 푧 ̄푦, and so acts on 푄 = 0. Re- étale groups. For example, when 픽푞 , 1 푛 #ℙ (픽 푛 ) = 푞 + 1 placing complex conjugation with the Frobenius involu- direct gives 푞 , and 푝 tion 푥 ↦ ̄푥= 푥 (special to characteristic 푝) of 픽푝2 over 1 픽푝 gives rise to an action of PGU3(픽푝2 ) on this curve. 푍ℙ1 (푇) = . (1 − 푞푇)(1 − 푇) Example 3: Fundamental groups. An important objective of mathematics is to classify spaces; a natural approach to In 1954 Lang and Weil [7] proved a weaker version of this is introducing and comparing algebraic invariants as- the Weil Conjectures: if 푋 has 푑 irreducible components sociated to a . One important example of such an of maximal 푟, then is the , which captures infor- 푛 푟 푟−1/2 #푋(픽푞푛 ) ∼ 푑(푞 ) + 푂(푞 ), mation about the geometry of a space and its maps to other spaces. For a variety 푋 over ℂ, the topological fundamen- as 푛 goes to infinity. These Lang–Weil estimates are useful tal group 휋1(푋) has a description in terms of loops on in both directions: information either about point counts 1 푋. In particular, the 픸 has trivial 휋1, since it’s con- or about the parameters 푟 and 푑 can be traded for the ℂ tractible. Using an equivalent description of 휋1, this says other. This technique is particularly striking in combina- 1 that 픸 has no nontrivial unramified covers. The same tion with the technique of spreading out from characteris- ℂ is true when we replace ℂ with any other characteristic 0 tic 0 to characteristic 푝: counting points on a specializa- . In characteristic 푝, the theory of étale covers gives tion to characteristic 푝 can give information about dimen- a direct analog 휋ét of the topological fundamental group. sion and irreducibility in characteristic 0! 1 Grothendieck proved that the prime-to-푝 part of 휋ét is the This idea was used in recent work of Browning–Vishe 1 same as that of 휋 of an analogous curve over ℂ [6]. How- [2], building on an idea of Ellenberg–Venkatesh, to show 1 ever, the 푝-part of 휋ét detects that the theory of covers is that certain spaces parameterizing rational curves on hy- 1 much richer in this setting. For example, over a character- persurfaces over ℂ are irreducible of the expected dimen- istic 푝 , 휋ét(픸1) is far from trivial; in fact, its sion. This is a beautiful illustration of the synergy between 1 cardinality is huge and depends on the ground field. It characteristic 0 and characteristic 푝 algebraic geometry. ét 1 is even conjectured that 휋1 (픸푘) determines the ground 푘 Characteristic 푝 phenomena. field when is algebraically closed. Example 1: Elliptic curves of unbounded rank. Another no- Mathematics research communities. Arithmetic geome- table application of these modern tools is an explicit ex- try in characteristic 푝 lends itself to a rich and varied col- ample, due to Ulmer [10], of a family of elliptic curves lection of accessible problems, in topics such as isogeny of unbounded rank defined over a function field over afi- classes of abelian varieties over finite fields, Galois cov- nite field. The construction makes clever use of the known ers of curves and lifting problems, and arithmetic dynam- cases of the [9] over finite fields. The exis- ics. Many of these problems are existential, and can be tence of such a family of elliptic curves over number fields attacked by an explicit or computational approach. Signif- is a topic of heated debate amongst number theorists to- icant progress in this setting has been made in the last year day! Numerous heuristics have been developed about the alone, due to both recent theoretical technical innovations

240 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 66, NUMBER 2 and to recent computational advances that have aided ex- perimentation. Relatedly, there are new approaches to ex- plicitly constructing examples exhibiting certain phenom- ena. We invite early-career mathematicians from a wide range of backgrounds to continue this story at our upcoming MRC, “Explicit Methods in Arithmetic Geometry in Char- acteristic 푝.”

References [1] E. Artin. Quadratische Körper im Gebiete der höheren Kongruenzen. I, II. Math. Z., 19(1):153–246, 1924. [2] Tim Browning and Pankaj Vishe. Rational curves on smooth hypersurfaces of low degree. Number The- ory, 11(7):1657–1675, 2017. [3] . La conjecture de Weil. I. Inst. Hautes Études Sci. Publ. Math., (43):273–307, 1974. [4] Pierre Deligne. La conjecture de Weil. II. Inst. Hautes Études Sci. Publ. Math., (52):137–252, 1980. [5] . Formule de Lefschetz et ratio- nalité des fonctions 퐿. In Séminaire Bourbaki, Vol. 9, pages Exp. No. 279, 41–55. Soc. Math. France, Paris, 1995. [6] Alexander Grothendieck. Revêtements étales et groupe fon- damental. Fasc. I, II, volume 1960/61 of Séminaire de Géométrie Algébrique. Institut des Hautes Études Scien- tifiques, Paris, 1963. [7] and André Weil. Number of points of varieties in finite fields. Amer. J. Math., 76:819–827, 1954. [8] . The Riemann Hypothesis in Char- acteristic 푝 in Historical Perspective. available at https://www.mathi.uni-heidelberg.de/ ~roquette/2017rv_with_pictures.pdf, 2018. [9] . Conjectures on algebraic cycles in 푙-adic coho- mology. In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 71–83. Amer. Math. Soc., Provi- dence, RI, 1994. [10] Douglas Ulmer. Elliptic curves with large rank over func- Credits tion fields. Ann. of Math. (2), 155(1):295–315, 2002. Photos of Renee Bell, Padmavathi Srinivasan, and Isabel Vogt [11] André Weil. Foundations of Algebraic Geometry. American are courtesy of Joe Rabinoff. Mathematical Society Colloquium Publications, vol. 29. Photo of Julia Hartmann is courtesy of Steven Kasich. American Mathematical Society, New York, 1946. Photo of Valentijn Karemaker is courtesy of Pasfoto Utrecht [12] André Weil. Sur les courbes algébriques et les variétés qui s’en Centrum. déduisent. Actualités Sci. Ind., no. 1041 = Publ. Inst. Math. Univ. Strasbourg 7 (1945). Hermann et Cie., Paris, 1948. [13] André Weil. Numbers of solutions of in finite fields. Bull. Amer. Math. Soc., 55:497–508, 1949.