Thinking Positive: ArithmeticGeometry in Characteristic 𝑝Renee Bell, Julia Hartmann, Valentijn Karemaker,

Padmavathi Srinivasan, and Isabel Vogt

The authors of this piece are organizers of the AMS2019 Mathematics Research Communities summerconference on Explicit Methods in Arithmetic

Geometry in Characteristic p, one of three topicalresearch conferences offered this year that are focused

on collaborative research and professionaldevelopment for early career mathematicians.

Additional information can be found athttp://www.ams.org/programs/research-

communities/2019MRC-Explicit.Applications are open until February 15, 2019.

Arithmetic geometry arose as a beautiful and powerful the-ory unifying geometry and number theory, formalizingstriking analogies between them in a way that allowedtools, results, and intuition of each to be transported tothe other—a notable example of this is the proof of thecenturies-old problem, Fermat’s Last Theorem. This the-ory provides a geometric viewpoint of objects over fields

Renee Bell is a Hans Rademacher Instructor at the University of Pennsylvania. Her email address is rhbell@math.upenn.edu.Julia Hartmann is a professor at the University of Pennsylvania. Her email address is hartmann@math.upenn.edu.Valentijn Karemaker is a postdoctoral research fellow at the University of Penn-sylvania. Her email address is vkarem@math.upenn.edu.Padmavathi Srinivasan is a Hale Visiting Assistant Professor at the Georgia Institute of Technology. Her email address is padmavathi.srinivasan @math.gatech.edu.Isabel Vogt is a Massachusetts Institute of Technology graduate student currently visiting Stanford University. Her email address is ivogt@mit.edu.For permission to reprint this article, please contact:reprint-permission@ams.org.

DOI: https://doi.org/10.1090/noti1804

of prime characteristic 𝑝, like finite fields. Decades afterthese ideas were formalized, characteristic 𝑝 arithmetic ge-ometry is rapidly expanding to include work in the vibrantyoung fields of arithmetic dynamics and derived algebraicgeometry.

Fields of positive characteristic 𝑝 have a fundamentallydistinct flavor from the classical setting of fields of charac-teristic 0. In addition, having to let go of the archimedeanframework requires a radically different geometric intuition.Working in characteristic𝑝 therefore comeswith additionalchallenges, but there are also additional tools and struc-ture that can be exploited, which have led to remarkableresults. In some instances these results even carry over tosolve open problems in characteristic 0.

Historical overview. While number theorists had beenstudying the rational numbers and other number fields us-ing the Riemann and Dedekind zeta functions, E. Artin inhis 1921 thesis [1] first proposed an analogous theory ofzeta functions for curves over finite fields. Hasse extendedthis to prove the Riemann hypothesis for elliptic curvesover finite fields. Their work further sparked the develop-ment of the theory of function fields, as well as the theoryof algebraic varieties, by several mathematicians, includ-ing Weil. In the 1940’s, Weil published a book [11] onthe foundations of algebraic geometry in characteristic 𝑝,and subsequently proved the Riemann hypothesis for func-tion fields over finite fields [12]. Moreover, in 1949, Weilproposed conjectures on the behavior of zeta functions ofvarieties over finite fields—including the Riemannhypothesis—which came to be known as the Weil conjec-tures [13]. The breakthroughwhich enabledGrothendieck,

FEBRUARY 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 239

M. Artin, and Verdier to prove all but one of the Weil con-jectures in 1964 [5] was their development of ℓ-adic coho-mology. Building on this, the Riemann hypothesis waseventually proven in two different ways by Deligne ([3,1974], [4, 1980]). For more historical details, see e.g. [8].

Weil conjectures and counting points. The zeta function𝑍𝑋(𝑇) of an algebraic variety 𝑋 over 𝔽𝑞 encompasses in-formation about point counts over all finite extensions of𝔽𝑞:

𝑍𝑋(𝑇) = exp(∑𝑛≥0

#𝑋(𝔽𝑞𝑛)𝑇𝑛

𝑛 ) .

The Weil conjectures predict that the power series 𝑍𝑋(𝑇)is a rational function, whose zeros and poles can be de-scribed in terms of natural group actions on the associatedétale cohomology groups. For example, when 𝑋 = ℙ1

𝔽𝑞 ,

direct calculation gives #ℙ1(𝔽𝑞𝑛) = 𝑞𝑛 + 1, and

𝑍ℙ1(𝑇) = 1(1 − 𝑞𝑇)(1 −𝑇).

In 1954 Lang and Weil [7] proved a weaker version ofthe Weil Conjectures: if 𝑋 has 𝑑 irreducible componentsof maximal dimension 𝑟, then

#𝑋(𝔽𝑞𝑛) ∼ 𝑑(𝑞𝑛)𝑟 +𝑂(𝑞𝑟−1/2),

as 𝑛 goes to infinity. These Lang–Weil estimates are usefulin both directions: information either about point countsor about the parameters 𝑟 and 𝑑 can be traded for theother. This technique is particularly striking in combina-tion with the technique of spreading out from characteris-tic 0 to characteristic 𝑝: counting points on a specializa-tion to characteristic 𝑝 can give information about dimen-sion and irreducibility in characteristic 0!

This idea was used in recent work of Browning–Vishe[2], building on an idea of Ellenberg–Venkatesh, to showthat certain spaces parameterizing rational curves on hy-persurfaces over ℂ are irreducible of the expected dimen-sion. This is a beautiful illustration of the synergy betweencharacteristic 0 and characteristic 𝑝 algebraic geometry.

Characteristic 𝑝 phenomena.

Example 1: Elliptic curves of unbounded rank. Another no-table application of these modern tools is an explicit ex-ample, due to Ulmer [10], of a family of elliptic curvesof unbounded rank defined over a function field over a fi-nite field. The construction makes clever use of the knowncases of the Tate conjecture [9] over finite fields. The exis-tence of such a family of elliptic curves over number fieldsis a topic of heated debate amongst number theorists to-day! Numerous heuristics have been developed about the

behavior of ranks and other fundamental invariants of el-liptic curves.

Example 2: Automorphisms. A broad theme in mathemat-ics is that one should study the symmetries (i.e. automor-phisms) of an object alongside the object itself. In the sim-plest case of an algebraic curve over ℂ, the automorphismgroup varies by the topological type: it is infinite for curvesof genus 𝑔 = 0 or 1, but once 𝑔 ≥ 2, Hurwitz famouslyproved that its order is at most 84(𝑔 − 1). In character-istic 𝑝, an algebraic curve could have additional symme-tries! For example, the projective plane curve 𝑥𝑝+1+𝑦𝑧𝑝+𝑧𝑦𝑝 over 𝔽𝑝2 has genus 𝑝(𝑝 − 1)/2, but 𝑂(𝑝8) symme-tries. These symmetries arise in analogy with the fact thatthe unitary group PGU3(ℂ) preserves the Hermitian form𝑄(𝑥,𝑦, 𝑧) = 𝑥 ̄𝑥 + 𝑦 ̄𝑧 + 𝑧�̄�, and so acts on 𝑄 = 0. Re-placing complex conjugation with the Frobenius involu-tion 𝑥 ↦ ̄𝑥 = 𝑥𝑝 (special to characteristic 𝑝) of 𝔽𝑝2 over𝔽𝑝 gives rise to an action of PGU3(𝔽𝑝2) on this curve.Example 3: Fundamental groups. An important objective ofmathematics is to classify spaces; a natural approach tothis is introducing and comparing algebraic invariants as-sociated to a space. One important example of such aninvariant is the fundamental group, which captures infor-mation about the geometry of a space and itsmaps to otherspaces. For a variety 𝑋 over ℂ, the topological fundamen-tal group 𝜋1(𝑋) has a description in terms of loops on𝑋. In particular, the line 𝔸1

ℂ has trivial 𝜋1, since it’s con-tractible. Using an equivalent description of 𝜋1, this saysthat 𝔸1

ℂ has no nontrivial unramified covers. The sameis true when we replace ℂ with any other characteristic 0field. In characteristic 𝑝, the theory of étale covers givesa direct analog 𝜋ét

1 of the topological fundamental group.Grothendieck proved that the prime-to-𝑝 part of𝜋ét

1 is thesame as that of𝜋1 of an analogous curve over ℂ [6]. How-ever, the 𝑝-part of 𝜋ét

1 detects that the theory of covers ismuch richer in this setting. For example, over a character-istic 𝑝 ground field, 𝜋ét

1 (𝔸1) is far from trivial; in fact, itscardinality is huge and depends on the ground field. Itis even conjectured that 𝜋ét

1 (𝔸1𝑘) determines the ground

field when 𝑘 is algebraically closed.

Mathematics research communities. Arithmetic geome-try in characteristic 𝑝 lends itself to a rich and varied col-lection of accessible problems, in topics such as isogenyclasses of abelian varieties over finite fields, Galois cov-ers of curves and lifting problems, and arithmetic dynam-ics. Many of these problems are existential, and can beattacked by an explicit or computational approach. Signif-icant progress in this setting has been made in the last yearalone, due to both recent theoretical technical innovations

240 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 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-careermathematicians fromawide rangeof backgrounds to continue this story at our upcomingMRC, “Explicit Methods in Arithmetic Geometry in Char-acteristic 𝑝.”

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. Algebra Number The-ory, 11(7):1657–1675, 2017.

[3] Pierre Deligne. La conjecture deWeil. I. Inst. Hautes ÉtudesSci. Publ. Math., (43):273–307, 1974.

[4] Pierre Deligne. La conjecture de Weil. II. Inst. HautesÉtudes Sci. Publ. Math., (52):137–252, 1980.

[5] Alexander Grothendieck. Formule de Lefschetz et ratio-nalité des fonctions 𝐿. In Séminaire Bourbaki, Vol. 9, pagesExp. 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 deGéométrie Algébrique. Institut des Hautes Études Scien-tifiques, Paris, 1963.

[7] Serge Lang and André Weil. Number of points of varietiesin finite fields. Amer. J. Math., 76:819–827, 1954.

[8] Peter Roquette. The Riemann Hypothesis in Char-acteristic 𝑝 in Historical Perspective. available athttps://www.mathi.uni-heidelberg.de/~roquette/2017rv_with_pictures.pdf, 2018.

[9] John Tate. Conjectures on algebraic cycles in 𝑙-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-tion fields. Ann. of Math. (2), 155(1):295–315, 2002.

[11] André Weil. Foundations of Algebraic Geometry. AmericanMathematical Society Colloquium Publications, vol. 29.American Mathematical Society, New York, 1946.

[12] AndréWeil. Sur les courbes algébriques et les variétés qui s’endéduisent. Actualités Sci. Ind., no. 1041 = Publ. Inst. Math.Univ. Strasbourg 7 (1945). Hermann et Cie., Paris, 1948.

[13] André Weil. Numbers of solutions of equations in finitefields. Bull. Amer. Math. Soc., 55:497–508, 1949.

Credits

Photos of Renee Bell, Padmavathi Srinivasan, and Isabel Vogtare courtesy of Joe Rabinoff.

Photo of Julia Hartmann is courtesy of Steven Kasich.Photo of Valentijn Karemaker is courtesy of Pasfoto Utrecht

Centrum.

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