Schedule and Abstracts

All talks will take place at the Institute of Mathematics. = Expository

Monday, August 13 9:00-9:10 Welcome words Gerhard Frey 9:10-10:40 Galois Theory: the Key to Numbers and Cyphers I Joachim von zur Gathen 11:00-12:45 Computational complexity and cryptography Lunch Break Roberto Avanzi 14:30-15:20 Arithmetic of supersingular Koblitz curves in characteristic three Joachim von zur Gathen 15:45-17:30 Computational complexity and cryptography

Tuesday, August 14 Gerhard Frey 9:00-10:45 Galois Theory: the Key to Numbers and Cyphers II Jim Massey 11:00-12:45 TBA Lunch Break Lenny Taelman 14:30-15:30 Cyclotomic function ﬁelds and values of L-functions Douglas Ulmer 15:45-16:45 Arithmetic of Jacobians in Artin-Schreier extensions Marc Palm 17:00-17:30 Which factors can occur in an automorphic representations?

1 Wednesday, August 15 Jim Massey 9:00-10:45 TBA Thomas Mittelholzer 11:00-12:00 Coding techniques for data storage systems Jan Tuitman 12:10-13:00 TBA Lunch Break Alexei Panchishkin 14:30-15:30 Applications of Drinfeld modules in cryptography Matthew Papanikolas 15:45-16:45 Explicit class ﬁeld theory for function ﬁelds Ralf Butenuth 17:00-18:00 Quaternionic Drinfeld modular forms and harmonic cocycles

Thursday, August 16 Federico Pellarin 9:00-10:00 On the values of Goss’ zeta function Alina C. Cojocaru 10:15-11:15 Frobenius distributions for generic Drinfeld modules of rank 2 Hans-Georg R¨uck 11:30-12:30 On uniformizable t-modules, an example Lunch Break Dinesh Thakur 14:30-15:30 Automata: Tools and appplications to function ﬁeld arithmetic Florian Breuer 15:45-16:45 An invitation to the Andre-Oort Conjecture Douglas Ulmer 17:00-18:00 Birch and Swinnerton-Dyer conjecture

2 Friday, August 17 Yuri Zarhin 9:00-10:00 Brauer-Grothendieck groups and Brauer-Manin sets Matthew Papanikolas 10:15-11:15 Hyperderivative power sums and log-algebraicity identities Michael Spiess 11:30-12:30 p-adic L-functions via group cohomology Lunch Break CécileArmana 14:30-15:30 Explicit bases of modular symbols over function fields Fu-Tsun Wei 15:45-16:45 On Rankin triple product L-functions and function field analogue of Gross-Kudla formula

Saturday, August 18 Dinesh Thakur 9:00-10:00 Higher multiplicities and arithmetic derivatives Florian Breuer 10:15-11:15 On Drinfeld modular polynomials Gebhard Böckle 11:30-12:30 Independence of `-adic Galois representations over finitely generated fields

3 Abstracts

CécileArmana (University of Münster) Explicit bases of modular symbols over function fields. We will discuss Teitelbaum’s theory of modular symbols over a rational function field of positive characteristic. They are an essential tool for computations of certain automorphic forms, Drinfeld modular forms and elliptic curves over such a field. As in the classical setting the space of modular symbols is described in a very combinatorial way by a finite presentation. We will explain how explicit bases may be derived for several congruences subgroups (such a statement is not known in the classical case). We will also mention applications.

Roberto Avanzi (Qualcomm Research) Arithmetic of supersingular Koblitz curves in characteristic three. In this presentation we show how knowledge from algebraic geometry, algebraic number theory and “low brow” fiddling with explicit aspects of the arithmetic of curves come together to bring significant practical performance enhancements. The main topic here are digital expansions of scalars for supersingular Koblitz curves in characteristic three – curves which can be used in pairing-based cryptography. Since in pairing-based protocols besides pairing computations also scalar multiplications are required, and the performance of the latter is not negligible, improving it is clearly important as well. The digital expansions are used in a kind of “double-and-add” scalar multiplication algorithm, where “doubling” is replaced by a the Frobenius endomorphism. The considered expansions of integers are therefore made not to an integral base, but to the algebraic base of τ, where τ is a zero of a polynomial τ 2 ± 3τ + 3. We consider several properties of these expansions and apply them to scalar multiplication on the aforementioned curves. For instance, we construct digit sets for these expansions as products of powers of algebraic integers associated to certain endomorphisms of the curve. The starting point is a structure theorem for unit groups of residue classes of a quadratic order associated to the Frobenius endomorphism. The algebraic integers we choose are associated to generators of these unit groups. There are of course several choices for these generators: we chose generators associated to endomorphisms for which we could find efficient explicit formulae in a suitable coordinate system. This is a joint work with Clemens Heuberger (Klagenfurt, Austria) and Helmut Prodinger (Stellenbosch, South Africa).

4 Gebhard Böckle (Heidelberg University) Independence of `-adic Galois representations over finitely generated fields. Let K be a finitely generated field (over its prime field) and let X be a scheme of finite type q over K. We consider the family of `-adic cohomology groups V` = Hc (X,Q`) while ` varies over the rational primes. For each ` this yields a Galois representation of the absolute Galois group GK of K on the finite dimensional vector spaces V`. Two basic questions are: (1) What is the image of GK in Aut(V`)? (2) Are the images independent? In the simplest case, K a number field and X an elliptic curve, both questions were answered by Serre in 1971. In general Question (1) is very difficult, already in the case that X is an abelian variety over K. The talk will focus on Question (2). It was recently answered by Serre if K is a number fields and by Gajda-Petersen if K is of characteristic zero. The talk is on joint work with W. Gajda and S. Petersen, which provides the “expected” answer for K of positive characteristic.

Florian Breuer (Stellenbosch University) On Drinfeld modular polynomials. We construct Drinfeld modular polynomials which relate the isomorphism invariants of rank r Drinfeld Fq[T ]-modules which are linked by an isogeny of a given type. We also prove a generalization of the Kronecker congruence relations for these poynomials. Joint work with Hans-Georg Rueck.

Ralf Butenuth (Heidelberg University) Quaternionic Drinfeld modular forms and harmonic cocycles. Drinfeld modular forms for congruence subgroups of GL2(Fq[T ]) were introduced in the 1980’s by Goss as an analogue for function ﬁelds of classical modular forms. By a result of Teitelbaum they are related to combinatorial objects, called harmonic cocycles, on a Bruhat- Tits tree. We study Drinfeld modular forms for inner forms on GL2 related to unit groups in maximal orders of quaternion algebras. We obtain a result analogous to Teitelbaum’s and give an explicit description of the space of harmonic cocycles, which is useful for computations.

Alina C. Cojocaru (University of Illinois at Chicago) Frobenius distributions for generic Drinfeld modules of rank 2. Let q be an odd prime power, A = Fq[T ], and k = Fq(T ). Let ψ be a generic Drinfeld A-module over k, of rank 2. We provide an explicit description of the Artin symbol in the division fields of psi, and discuss applications to problems focused on the splitting of primes of A in infinite families of division fields of ψ. This is joint work with Mihran Papikian.

5 Gerhard Frey (University of Duisburg-Essen) Galois Theory: the Key to Numbers and Cyphers. Very often Diophantine problems can be stated in an elementary way but it is notoriously hard to solve them. The most famous example for this phenomenon was Fermat’s Last Theorem. The situation becomes better whenever one ﬁnds a mathematical structure behind the problem, and in many cases this structure is delivered by the action of the Galois group on geometric objects like torsion points of elliptic curves or, more generally, abelian varieties. Then the arithmetic of Galois representations plays a dominant role. A key role in this game is occupied by Jacobian varieties of modular curves. These varieties are very well understood, and connections to modular forms allow deep theoretical and practical insights. A typical and very interesting aspect is that there is an intense interplay between theoretical and algorithmic approaches. And there is an amazing and important additional bonus: Many of the used methods and obtained results can be applied to construct public key crypto systems and to discuss their security. In the lectures we shall try to explain both background and results as well as conjectures and applications of this in this exciting part of arithmetic geometry.

Joachim von zur Gathen (Bonn-Aachen International Center for Information Technology) Computational complexity and cryptography. This lecture series exemplifies the influence of theoretical computer science, and of complexity theory in particular, on cryptography. The first example are reductions, coming from computability and the theory of P vs. NP, and now a standard tool in theoretical cryptography. The second one concerns structured models, called generic computations in cryptography. Here one does not consider a general model of computation like a Turing machine, but specialized calculations, such as arithmetic operations for arithmetic problems. In such a model, the discrete logarithm problem can actually be proved to be hard. The lectures end with an idiosyncratic view on ethics in science.

6 Thomas Mittelholzer (IBM Zurich Research Laboratory) Coding Techniques for Data Storage Systems. From a coding perspective, data storage devices can be considered as a communication channel in time, where data is stored at some instance in time and read back later. In addition to the underlying channel characteristics, coding for storage needs to take application speciﬁc requirements into account, namely, very low error rates, limited delay, and limited (decoding) complexity. After a brief introduction to algebraic coding theory, typical coding techniques for storage systems will be presented. A key aspect is the assessment of the code performance at very low error rates. For bounded distance decoding, the analysis is based on the minimum distance and weight enumerators of linear codes. For codes based on graphs - such as low-density parity-check codes, the analysis relies on graph-based objects such as stopping sets, which determine the low error-rate performance of graph-based decoders.

Matthew Papanikolas (Texas A&M University) Hyperderivative power sums and log-algebraicity identities. Power sums of polynomials over finite fields are central in the theory of characteristic p zeta values and L-values. Thakur showed that these sums can be used to discover precise log-algebraicity identities on the Carlitz module. In investigating log-algebraicity on tensor powers of the Carlitz module, we find that power sums of polynomials and their hyperderivatives play a key role. We will discuss both vanishing criteria and explicit formulas for hyperderivative power sums, as well as demonstrate their connections with log-algebraicity and special values of L-functions.

Federico Pellarin (University of Saint-Etienne) On the values of Goss’ zeta function. We can associate Goss zeta function to a ring A when it can be obtained as the integral closure of Fq[T ] in a ﬁnite extension of Fq(T ), yielding some analogies with Dedekind’s zeta function. In this talk, however, we will essentially discuss the case A = Fq[T ] itself. Then, a wide phenomenology (zeros, special values etc.) suggests a parallel with Riemann’s zeta function. In particular, denoting this function associated to A = Fq[T ] by ζ, we have the identity ζ(0) = 1, which is easily proved by the simplest combinatorial arguments. Since no known functional equation for this zeta exists, this identity cannot be deduced from analytic properties of some gamma factor. We will consider another class of (entire rigid analytic) functions, satisfying functional equations from which, for instance, the formula ζ(0) = 1 above can be deduced analytically. Other features of these functions will be also described.

7 Marc Palm (GöttingenUniversity) Which factors can occur in an automorphic representations? Modular forms, Maass forms, and more generally irreducible automorphic representations over a global field admit a factorization into local irreducible representations. A local representation is a unitary representation of GL2(F ) for a local field F . There are uncountably many local representations, but only countably many automorphic representations. It is a natural question to ask, which combinations of factors can occur in an automorphic representation. I will talk about one tool box (trace formulas), its limitations and discuss my results in the context of GL(2).

Hans-Georg R¨uck (University of Kassel) On uniformizable t-modules, an example. We study a family of t-modules and give suﬃcient and necessary conditions for their uniformizability.

Michael Spiess (Bielefeld University) p-adic L-functions via group cohomology. We introduce a construction of the p-adic L-function of a totally real ﬁeld via group cohomology and explain an application to the phenomena of trivial zeros.

Lenny Taelman (Leiden University) Cyclotomic function ﬁelds and values of L-functions. Let A = Fq[T ], let P ∈ A be monic irreducible. For a (non-trivial) character × × χ :(A/P A) → Fq one has the Goss L-value

X χ(a) L(1, χ) = ∈ ((1/T )) a Fq a (with a ranging over the monic a ∈ A that are coprime with P ), as well as a P - adic version of this L-value. In this talk we will give arithmetic interpretations of these L-values. In particular, we will consider characteristic p analogues of the Mazur-Wiles theorem and the Kummer-Vandiver conjecture. This is joint work with Bruno Angl´es.

8 Douglas Ulmer (Georgia Institute of Technology) Arithmetic of Jacobians in Artin-Schreier extensions. I will discuss a construction of Jacobians over function ﬁelds which satisfy the BSD conjecture and have large Mordell-Weil rank in towers of Artin-Schreier extensions. Much of this is parallel to previous results on Kummer extensions, but one remarkable new phenomenon occurs: height pairings on one elliptic curve are related to point counts on another family of elliptic curves, somewhat reminiscent of mirror symmetry. This is joint work with Rachel Pries.

Fu-Tsun Wei (National Tsing-Hua University) On Rankin triple product L-functions and function field analogue of Gross- Kudla formula. In this talk, I will discuss the analytic properties of triple product L-functions associated to Drinfeld type newforms whose levels are equal and square-free. Automorphic forms of Drinfeld type can be viewed as function field analogue of weight 2 modular forms. After a brief review of the basic facts about Drinfled type cusp forms, I will give the functional equation of these triple product L-functions. When the “root number” is positive, I present an analogue of Gross Kudla formula for the central critical values. Two examples will be shown at the end.

Yuri Zarhin (Pennsylvania State University) Brauer-Grothendieck groups and Brauer-Manin sets. We discuss ﬁniteness properties of Brauer-Grothendieck groups with special reference to abelian varieties and K3 surfaces. This is a report on a joint work with Alexei Skoroboga- tov.