L-functions and Motives in Niseko 2024

Toshiro Hiranouchi , Extended differential symbol and the Weil reciprocity law with modulus

For a field k of characteristic p>0, the etale mod p motivic cohomology H^i_L(k,Z/p(n)) of Spec(k) of weight n is concentrated in the degree i = n and n+1. According to the Bloch-Gabber-Kato theorem, the degree n-part H^n_L(k,Z/p(n)) is described by the n-th Milnor K-group of the field k via the differential symbol map. In this talk, we introduce a similar description for the degree (n+1)-part H^{n+1}_L(k,Z/p(n)) using an extended differential symbol map. By comparing our results with a "tensor product" in the category of reciprocity sheaves (Rülling-Sugiyama-Yamazaki/Koizumi-Miyazaki), we show that a Weil reciprocity law with modulus is killed by the Artin-Schreier map. This is a joint work with R. Sugiyama.

Tetsushi Ito, Arithmetic equivalence of elliptic curves over number fields

In 1926, Gassmann discovered non-isomorphic number fields having the same Dedekind zeta functions; such number fields are called "arithmetically equivalent." Since the work of Gassmann, the notion of arithmetical equivalence has been refined and generalized to several directions. In this talk, we study similar problems for elliptic curves. Namely, we study elliptic curves over non-isomorphic number fields having the same Hasse-Weil L-functions. We propose to call such curves "arithmetically equivalent." We give several examples of arithmetically equivalent elliptic curves. We also give a complete classification of arithmetically equivalent non-CM elliptic curves over non-isomorphic quadratic fields based on the results of Elkies and Ribet on "Q-curves." This talk is based on a joint work with Masataka Chida (Tokyo Denki University) and Sho Yoshikawa (Tokyo University of Science).

Ryomei Iwasa, Atiyah duality for motivic spectra

I’ll explain Atiyah duality established in the setting of non-A^1-invariant motivic spectra. This explains every known Poincaré duality for cohomology theories of schemes under a unified principle; examples include p-adic cohomology theories such as crystalline cohomology and prismatic cohomology. Furthermore, our version of Atiyah duality allows a good control of the “A^1-colocalization” of motivic spectra, which transforms a cohomology theory into an A^1-invariant cohomology theory without changing its values on smooth projective schemes; for example, Berthelot’s rigid cohomology is the A^1-colocalization of rational crystalline cohomology. Integrally, the A^1-colocalization of crystalline cohomology recovers the logarithmic crystalline cohomology of strict normal crossings compactifications, and shows that the latter is independent of the choice of compactifications, which has been a fundamental open question since the late 80’s. Joint work with Toni Annala and Marc Hoyois. 

Wataru Kai, Linear patterns of prime elements in number fields

I will discuss my recent result that gives a sufficient condition for a set of finitely many polynomials of degree 1 with coefficients in a number ring to attain simultaneous prime values. This extends a 2012 theorem of Green-Tao-Ziegler from the case of Z to the general case. Time permitting, I will mention how this can be applied to produce (modestly) new families of varieties over number fields which satisfy the Hasse principle for rational points by using the so-called fibration method.

Tomokazu Kashio, On periods and the absolute Frobenius on Fermat curves

Values at rational points of the Γ-function appear in periods of elliptic curves with complex multiplication and Fermat curves (Chowla-Selberg, Rohrlich). On the other hand, the action of the absolute Frobenius on these algebraic curves involves the p-adic Γ-function (Ogus, Coleman). Is this coincidence or something more fundamental? In this talk we explore the relation between monomial relations of CM-periods and functional equations that characterize the Γ-function and the p-adic Γ-function. As a result, we observe that the appearance of the p-adic Γ-functions in the absolute Frobenius is, in a sense, inevitable. We also introduce a conjectural generalization to arbitrary CM fields (Yoshida’s absolute CM-periods and its p-adic analogue).

Junnosuke Koizumi, Blow-up invariance of cohomology theories with modulus

In this talk, I will discuss cohomology theories for modulus pairs, that is, pairs consisting of a smooth algebraic variety and an effective divisor on it. In particular, I will present my recent result showing that well-behaved cohomology theories are automatically invariant under admissible blow-ups. This generalizes the blow-up invariance of Hodge cohomology established by Kelly-Miyazaki. If time permits, I will also touch upon the construction of a new category of motives based on the blow-up invariance.

Teruhisa Koshikawa, Cohomology of log prismatic F-crystals

Bhatt and Scholze introduced the prismatic cohomology, which is an integral p-adic cohomology thoery for schemes in mixed characteristic. Their theory and subsequent developement also provide a good theory of coefficients. I will talk about a logarithmic variant of this story, based on my previous works (partly with Zijian Yao) and ongoing work with Kentaro Inoue. 

Yoshinori Mishiba, On algebraic relations among values of multiple polylogarithms in positive characteristic

Let K be a rational function field in one variable over a finite field with a fixed infinite place, and let v be a finite place of K. The Carlitz multiple polylogarithms (CMPLs) over K are introduced by Chang as generalizations of the logarithmic function of the Carlitz module. Their infinity-adic values at (K-)algebraic points have period interpretations of certain t-motives, and it is considered important to study their properties. In this talk, we show that the v-adic values of CMPLs at algebraic points satisfy the same K-algebraic relations that the corresponding infinity-adic values of CMPLs satisfy. This is a joint work with Chieh-Yu Chang and Yen-Tsung Chen.

Kazuaki Miyatani, Cancellation theorem for p-adic hypergeometric D-modules

In this talk, I will discuss how arithmetic D-modules defined by hypergeometric equations (under a p-adic non-Liouvilleness condition) appear as extensions of other arithmetic D-modules. This work serves as a p-adic counterpart to a work of N. M. Katz for the complex and l-adic cases. It also provides a cancellation formula for hypergeometric functions over finite fields when the parameters are rational.

Atsuhira Nagano, Kummer-like surfaces and their partner surfaces

Kummer surfaces for principally polarized Abelian surfaces are K3 surfaces defined by quartic equations. They are important in algebraic geometry and number theory. Particular families of lattice polarized K3 surfaces naturally contain the family of Kummer surfaces. The speaker is expecting that such families, which he calls the families of Kummer-like surfaces, are also interesting objects of research. In this talk, we will see properties of Kummer-like surfaces, such as lattices, periods, elliptic fibrations or defining equations. In order to study them, the speaker will introduce useful K3 surfaces, which he calls partner surfaces. Moreover, if there is enough time, the speaker will show the relation among K3 surfaces studied by Matsumoto-Sasaki-Yoshida (1992), Clingher-Doran (2012), Clingher-Malmendier-Shask (2019), N. (2021) and N.-Shiga (2023).

Akio Nakagawa, Geometric interpretation of certain transformation formulas for  hypergeometric functions over finite fields

Over the complex numbers, the Gauss hypergeometric function, the Kummer hypergeometric function and their generalizations have been studied for a long time, and their analogous functions over finite fields have been studied since the 1980s. In this talk, first, we will see that a hypergeometric function over a finite field is the number of rational points on a subvariety of a product of Fermat hypersurfaces and Artin-Schreier curves over the finite field. Secondly, we will give a geometric interpretation of certain formulas for hypergeometric functions over the finite field by constructing isomorphisms between the subvarieties.

Yoshiyasu Ozeki, Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields

It is a theorem of Ribet that an abelian variety defined over a number field K has only finitely many torsion points with values in the maximal cyclotomic extension field K^cyc of K. Recently, Rössler and Szamuely generalized Ribet's theorem in terms of the étale cohomology with Q/Z-coefficients of a smooth proper variety. In this talk, I will explain that the same finiteness holds even after replacing K^cyc with the field obtained by adjoining to K all roots of all elements of a certain subset of K. Furthermore, I give some new examples of TKND-AVKF fields; the notion of TKND-AVKF is introduced by Hoshi, Mochizuki and Tsujimura, and TKND-AVKF fields are expected as one of suitable base fields for anabelian geometry. This is a joint work with Takahiro Murotani. 

Kazuma Shimomoto, Perfectoid towers, Frobenius lifts and lim Cohen-Macaulay sequences

I will talk about several ways of constructing non-regular Noetherian rings of mixed characteristic with Frobenius lifts. The resulting rings have some interesting singularities that were not known previously.  Then I will talk about their connections with perfectoid towers and potential applications to an open problem in commutative algebra. This is a report on a joint work with S. Ishiro. R. Ishizuka and K. Nakazato.

Daichi Takeuchi, Constructing a nearby cycles functor for functions on mixed characteristic schemes

TBA

Place : Setsu Niseko, Ground Floor, "Park 90"

• Ryo Ishizuka (Tokyo Institute of Technology), Frobenius maps on mixed characteristic rings via prismatic cohomology

• Yusuke Nemoto (Chiba University), Non-torsion algebraic cycles on the Jacobians of Fermat quotients 

• Ryo Ohkawa (Osaka Metropolitan University), K-theoretic wall-crossing formulas and multiple basic hypergeometric series

• Asuka Shiga (Tohoku University), Tate-Shafarevich group analogy of genus theory via Poitou-Tate duality and 2-descent