We present a framework for encoding information about objects from higher arithmetic geometry. This framework is built around a new kind of data type called a Tannakian symbol. The arithmetic objects we have in mind include modular forms (and more general automorphic representations), elliptic curves (and more general schemes, motives and algebraic stacks), nite graphs, group representations, and multiplicative functions (like the Euler totient function). The language of Tannakian symbols not only allows for representations of individual objects, but also representations of classes of objects, relations between objects, and various important unary and binary operations on objects. The development of this framework is the rst small step in a long-term project aiming to apply machine-learning algorithms to some problems of current interest in modern arithmetic geometry.
Arithmetic geometry is one of the most vibrant and abstract areas of modern pure mathematics. Out of the seven Millennium Problems, four come from pure mathematics, and of these four, one is solved and the remaining three belong to arithmetic geometry.
The prospect of arti cially intelligent programs making new and deep
discoveries in this area of mathematics is a tantalizing one. However, most of the
concepts encountered in modern arithmetic geometry are not easily stored or
manipulated by a computer. In the very long term, one may hope that advances
in mathematical linguistics, as developed in Ganesalingam's thesis [
In this project paper, we present a framework for encoding data about
objects from arithmetic geometry, with the aim of laying the foundation for future
applications of machine-learning techniques in the eld. We emphasize that this
is a rst brief survey of a long-term project, focussing on examples. A more
detailed discussion of potential applications will be provided in future publications
and in discussions at the CICM conference, but as a rst application, we
propose experiments with automated conjecture-making on invariants of motives
and schemes, using the language of Tannakian symbols presented here, together
with the SAGE package for domain-independent automated conjecture-making
developed recently by Larson and Van Cleemput [
Somewhat informally, we shall use the term \arithmetic object" to refer to any kind of object that is of central importance in arithmetic geometry. Some classes of such objects are: (1) Geometric objects (e.g. a scheme). The reader unfamiliar with the theory of schemes may think of a scheme simply as a system of polynomial equations. (2) Algebraic objects (a group, a ring, a Hopf algebra, etc.). (3) Homotopical objects (like an algebraic stack or a ring spectrum). (4) Combinatorial objects (for example a graph). (5) Analytic objects (e.g. a zeta function). (6) Objects in a Tannakian category. Examples of the latter include representations of nite groups, representations of Lie groups, Galois representations, automorphic representations, motives, Hodge structures, and F-isocrystals.
Technical de nitions of all the above terms (schemes, Tannakian categories,
etc) can be found in the online Encyclopedia of Mathematics [
In addition to seeking encodings of single objects, a central goal of our work is to also encode information about classes of objects (for example the class of objects in some given Tannakian category), operations on objects (such as tensor product of group representations, or Tate twist of motives, or Dirichlet convolution of multiplicative functions), relations between objects (such as a representation being a direct summand of another), and invariants of objects (like the Euler characteristic of a scheme). 1.2
Let C be some class of arithmetic objects, for examples the class of all elliptic curves over the rational numbers, or the class of all nite undirected graphs, or the class of all complex representations of the Monster group.
We seek an encoding framework for objects in C satisfying the following properties: 1. To every object X in the class C we can assign a nite amount of structured data E(X). (We think of E(X) as an elementary or electronic "shadow" of the object X.) 2. Given a description of X, there should be an explicit algorithm computing E(X). 3. Many important invariants of X should be computable from E(X) only. 4. Many important operations on objects in C should correspond to explicit manipulations of the corresponding structured data. 5. Given two objects X and X0 from di erent classes (say one graph and one elliptic curve), the two associated pieces of data E(X) and E(X0) should "be of a similar form" (to facilitate the discovery of connections between di erent kinds of structures). 6. Many of the deepest theorems and conjectures about objects X in modern arithmetic geometry should have a formulation in terms of the associated data E(X) only.
Any encoding satisfying these requirements will have the property that a computer could in principle discover (or guess) interesting mathematical statements by searching for patterns in the structured data of many arithmetic objects.
We have found an approach that satis es all of the above criteria for many classes of arithmetic objects. The framework is built around the notion of a Tannakian symbol. In many cases, the information contained in the Tannakian symbol is the same as the information contained in the \zeta function" of the arithmetic object, but Tannakian symbols are more exible than zeta functions, and there are also cases where it makes sense to speak of Tannakian symbols even though there are no zeta functions around. 1.3
We are not aware of any previous work with the explicit ambition of
applying machine-learning algorithms to geometric, Tannakian and homotopical
categories in higher arithmetic geometry. However, we have drawn inspiration from
many places. Due to algorithmic breakthroughs over the past decade by
Kedlaya [
The aim of this section is to de ne what Tannakian symbols are, and to summarize their most important algebraic properties. Proofs of these statements will be given elsewhere.
We begin by recalling some de nitions from abstract algebra. A monoid is a set equipped with a binary operation that is associative and has an identity element. A group is a monoid in which each element has an inverse. A monoid is called commutative if its binary operation is commutative. An abelian group is the same thing as a commutative group.
Example 1. The set of positive integers N is a monoid under addition, and it is also a monoid under multiplication. The set of all complex roots of unity is a monoid under multiplication.
A commutative ring is a set R with two binary operations, called addition (+) and multiplication ( ), with the requirements that R is an abelian group under addition, a commutative monoid under multiplication, and multiplication distributes over addition. The identity element for addition is denoted by 0, and the identity element for multiplication is denoted by 1.
Example 2. The set of integers Z is a commutative ring. The set Z=m, identi ed with f0; 1; : : : ; m 1g is a commutative ring for any integer m 2, in which addition and multiplication are carried out modulo m. Whenever R is a commutative ring, the set R[x] of polynomials in x with coe cients in R is also a commutative ring.
A eld is a commutative ring in which the nonzero elements under multiplication form a group (and not just a monoid).
Example 3. The set Q of rational numbers is a eld, and so is the set R of real numbers, and the set C of complex numbers.
A monoid homomorphism from one monoid to another monoid is a function which commutes with the binary operation and sends the identity element to the identity element. A ring homomorphism from a ring to another ring is a function which is a monoid homomorphism both with respect to addition and with respect to multiplication. An isomorphism (of rings or of monoids) is a homomorphism which admits a two-sided inverse.
Example 4. Let q be a positive integer. It is known that there exists a eld with exactly q elements if an only if q is a power of a prime number (i.e. q = pe for some prime p and some positive integer e). Two such nite elds with the same number of elements are always isomorphic (i.e. there exists an ring isomorphism between them), and we write Fq for any nite eld with exactly q elements.
It is possible to describe all nite elds in a very concrete way. First of all, when q is a prime number, the ring Z=q is a eld with q elements. A more interesting example is the eld with four elements F4, which can be described as the set f0; 1; ; + 1g where addition is carried out modulo 2, and multiplication is carried out modulo 2 and modulo the relation 2 = + 1. Similar models of nite elds exist (but are not in general unique) for any prime power q.
A lambda-ring is, informally, a commutative ring R \equipped with all possible symmetric operations". The precise de nition of \all possible symmetric operations" is expressed in the notion of a lambda-structure on a commutative ring. The general de nition of \lambda-structure" is given in terms of an in nite sequence 0, 1, 2, . . . of functions (not ring homomorphisms!) from R to R, satisfying axioms that are a bit complicated. However, when the ring R is torsion-free (meaning that nite sums x + x + : : : + x are never zero unless x itself is zero), there is a simpler equivalent de nition which we give here. All lambda-rings in this paper will be torsion-free, so this de nition is enough for our purposes.
De nition 1. Let R be a torsion-free commutative ring. A lambda-structure on R is an in nite sequence of ring homomorphisms 1, 2, . . . from R to R satisfying the following axioms: 1. 2. 3.
1(x) = x for all x 2 R. m( n(x)) = mn(x) for all m; n and all x 2 R.
p(x) xp (mod pR) for all prime numbers p and all x 2 R.
tiple of p, in the ring R. The homomorphisms p(x) xp can be written as a mulm are called Adams operations. 2.2
The kind of \structured data" we shall construct (denoted by E(X) in the introduction) will be called a U -indexed M -valued Tannakian symbol, and we now turn to the explanation of what this means.
Recall that a multiset is a unordered collection of elements, in which elements are allowed to be equal. For example, f2; 2; 2; 5g is a multiset with four elements taken from the set of integers.
De nition 2. Let M be a monoid. An M-valued Tannakian symbol is an ordered pair (A; B) of disjoint nite multisets with elements taken from M . We write TS(M ) for the set of all M -valued Tannakian symbols.
Conventions: We shall use the notation A=B or BA for the ordered pair (A; B), and will refer to A as the upstairs multiset and to B as the downstairs multiset. Also, if M happens to be a ring and we write TS(M ), we always think of M as a multiplicative monoid (in other words, we forget the additive structure). Example 5. The symbol f1; 1; i; 1; i =
g ; is an example of a C-valued Tannakian symbol. Here i is a complex square root of -1 and ; is the empty multiset. De nition 3. Let U be a set. A U -indexed M -valued Tannakian symbol is a function from U to TS(M ). The set of U -indexed M -valued Tannakian symbols will be denoted by TSU (M ).
Example 6. Let P be the set of prime numbers, and let p denote a variable element of P. Then fp2; 1g=fp; pg is a P-indexed N-valued Tannakian symbol.
Consider multisets A = fa1; a2; : : :g, B = fb1; b2; : : :g, C = fc1; c2; : : :g and D = fd1; d2; : : :g where all the elements are taken from the same monoid M . We de ne operations on Tannakian symbols by the following formulas: (Here ] denotes disjoint union of multisets.)
A
B A B
C D
C D = (Here, if the element a is repeated several times in A, the element an is also repeated the same number of times on the right hand side.) In each of these operations, it is understood that if the operation results in a symbol in which the upstairs and the downstairs multisets are not disjoint, then we remove pairs of identical elements until the multisets are disjoint. A few examples will illustrate what this means.
Example 7. Computations in TS(Z): Theorem 1. For any monoid M , the set TS(M ) is a lambda-ring under the operations , and n. The same is true for TSU (M ) for any set U . Furthermore, T SU (M ) is functorial in M as well as in U .
One can go on and give explicit de nitions of other structural features and invariants of Tannakian symbols, such as exterior powers, symmetric powers, virtual dimension, super-dimension, supertrace and superdeterminant. All this terminology comes from the setting of lambda-rings obtained by decategorifying Tannakian categories, but is retained also in situations where there is no Tannakian category involved. The point of all this structure is that whenever elements of the monoid M can be stored and manipulated by a computer, the same is true for elements of TSU (M ), and the latter capture huge amounts of structure relevant for higher arithmetic geometry.
Now we can be a bit more precise about the picture we would like to paint of structured data assigned to arithmetic objects. Return to the situation where C is some class of arithmetic objects. Given such a class, we can in many cases choose a monoid M and a set U and construct a map
E : C ! TSU (M ) which satis es most of the requirements in the introduction.
In such a setup, we are interested in the following general goals. 1. Understand how much information is lost when we pass from X to E(X). A way of making this more precise is to de ne the ber of a Tannakian symbol S as the set of all arithmetic objects X in C with E(X) = S. In many cases one can either prove that each ber consists of at most one element, or give a bound on the size of the ber. 2. Describe the image of E. 3. Set up a correspondence between operations on arithmetic objects in C and operations on Tannakian symbols. 2.4
Let a0; a1; a2; : : : be a linearly recursive sequence in C, with a0 = 1. It is wellknown that it is then possible to rewrite the power series a0 + a1t + a2t2 + : : : as a rational expression of the form Qjn=1(1 j t)= Qim=1(1 it), and we de ne the Tannakian symbol attached to the linearly recursive sequence to be A=B, with the multisets A = f ig and B = f j g. With this de nition, taking the product of power series corresponds to adding Tannakian symbols. 3 3.1
For the purposes of this article, we de ne an a ne scheme to be a nite set of variables x1; x2; : : : ; xd together with a nite list of polynomial equations (with integer coe cients) in these variables. Given an a ne scheme X and a eld K, we write X(K) for the set of solutions to the equations of X with values in the eld K; elements of this set are called K-valued points of X.
We also de ne a projective scheme to be a nite set of variables x0; x1; : : : ; xd together with a nite list of homogeneous polynomial equations (with integer coe cients) in these variables. The equations are not required to be of the same degree. In this setting, we let X(K) denote the set of equivalence classes of solutions with values in K, where two solutions are called equivalent if one is a scalar multiple of the other. For projective schemes, we never count the trivial solution in which all variables take the value zero.
As a special case we may take K to be the nite eld Fq, and the set X(Fq) is then automatically a nite set. We write #X(Fq) for the cardinality of this set.
There is a general construction which associates a projective scheme to any a ne scheme. If the a ne scheme X is de ned by equations fi(x1; x2; : : : ; xd) = 0 i = 1; 2; : : : m then the associated projective scheme is de ned by corresponding equations Fi(x0; x1; x2; : : : ; xd) = 0 i = 1; 2; : : : m where Fi is obtained from fi by multiplying each term by a suitable power of x0 so that Fi becomes homogenous of degree deg(fi).
Example 8. The equation x2 +1 = 0 de nes an a ne scheme X (in one variable). It is easy to see that in this case, we have X(Q) = X(R) = ; (the empty set), but X(C) = fi; ig. Using modular arithmetic, we compute X(F2) = X(F3) = ; and X(F5) = f2; 3g. In general, for an odd prime p, the cardinality of X(Fp) is 2 or 0 depending on whether p is congruent to 1 or 3 modulo 4. This pattern is a special case of Gauss' famous quadratic reciprocity law, and quadratic reciprocity is an example of a pattern that can be expressed purely in terms of Tannakian symbols.
Theorem 2 (Dwork). Let X be a scheme (a ne or projective) and let p be a prime. There exists unique multisets A = f 1; 2; : : : ; mg and B = f 1; : : : ; ng of complex numbers such that for all k 1, we have #X(Fpk ) = 1k + 2k + : : : + k n k 1 k 2 : : : k m De nition 4. Let X be a scheme and let p be a prime. We de ne the Tannakian symbol of X at p to be A=B, where A and B are the multisets in Dwork's theorem. Example 9. Since Wiles proved Fermat's Last Theorem using the Modularity theorem for elliptic curves, the class of elliptic curves has probably become the most famous class of schemes. As a simple example of an elliptic curve, take the scheme X de ned by the equation y2 + y = x3 x2. At the prime p = 2, the symbol becomes: f 1 + i; 1 f2g
ig A ne case:
f 1 + i; 1 f1; 2g ig The projective case is often the most interesting. In this example, deleting all numbers in the symbol except those with absolute value p2 corresponds to cutting out the \motive" h1(X) from X. One can also associate Tannakian symbols to modular forms, and the Modularity theorem can be formulated as saying that for every elliptic curve X, there exists a modular form which at all primes has the same Tannakian symbol as the motive h1(X). Combining Tannakian symbols from all primes gives rise to a map from the class of elliptic curves to TSP(C). The bers of this assignment are called isogeny classes of elliptic curves; it is known that these bers are nite. Furthermore, elliptic curves come with a natural complexity measure N called the conductor (the above example has conductor 11), and by restricting attention to elliptic curves of, say, conductor less than 100000, we may restrict the set of indexing primes to a nite set without losing any information.
In general, the symbol attached to a projective scheme without singularities yields easy recipes for computing the Betti numbers and Euler characteristic of a scheme. In the above example the Betti numbers are 1, 2 and 1; these numbers are obtained by counting symbol elements with absolute value 1, p2, and 2, respectively. The Euler characteristic is computed by subtracting the number of elements upstairs from the number of elements downstairs; for this elliptic curve we get 2 2 = 0. Plotting the numbers appearing in the symbol as points in the complex plane reveals symmetries related to Poincare duality and patterns related to the Riemann hypothesis over nite elds (proved by Deligne, Fields medal 1978). All of this was originally formulated as the famous Weil conjectures in the 1950s.
After computing the Tannakian symbols for several primes (up to size pN
approximately) one can easily compute what's called values of L-functions - these
are the values appearing in the two Millennium Problems called the (global)
Riemann hypothesis and the Birch and Swinnerton-Dyer conjecture. These
Tannakian symbols also allows for explicit formulations of many other deep questions
of current interest to arithmetic geometers, such as the Sato-Tate conjecture,
and various conjectures on Galois representations. The operations on Tannakian
symbols correspond to operations in the so-called Grothendieck ring of motives,
which is of interest not only in arithmetic geometry, but also in physics, where
they are directly related to Feynman integral calculations in perturbative
quantum eld theory [
be the "quartic Dwork family", i.e. the projective scheme de ned by the Let X equation
x4 + y4 + z4 + w4 = 4 xyzw where is a integer-valued parameter (so that by varying we get a family of schemes). The scheme X comes with a natural action of the group Z=4 Z=4. Taking the quotient scheme by this group action and resolving singularities yields a new scheme Y , called the mirror of X .
For concreteness, let's look at the prime p = 41. For = 2, we get1 the
following symbols:
E(X2) = f1; 41; 41; 41; 41; 41; 41; : : : ; 41;
1 The examples here are adapted from the presentation of Ursula Whitcher [
25 8p66i 25 + 8p66i
E(Y2) = f1; 41; 41; : : : ; 41; 41; 2 ; 2 ; 1681g=; with 19 copies of the number 41, a single copy of the number -41, and an otherwise identical symbol!
Still working with p = 41, for the case = 3 we get: E(X3) = f1; 41; 41; : : : ; 41; 39 + 4p10i; 39 4p10i; 1681g=; with 20 copies of the number 41. And this time, the Tannakian symbol for the mirror variety Y3 is
E(Y3) = f1; 41; 41; : : : ; 41; 39 + 4p10i; 39 4p10i; 1681g=; with 20 copies of 41, which means. . . that the symbols are absolutely identical!!
Patterns of this kind is the subject of arithmetic mirror symmetry, a relatively recent eld inspired by the physics of string theory. It is conceivable that a computer searching for patterns in Tannakian symbols could have identi ed the schemes X and Y as "similar", even if no human had ever thought of mirror symmetry. 4 4.1
Much of elementary number theory (questions about primes, divisibility, etc.), can be formulated in terms of multiplicative functions from N to C. In this context a function f is multiplicative if f (1) = 1 and f (mn) = f (m)f (n) whenever m and n are coprime.
Let p be a prime. For all multiplicative functions appearing naturally in number theory, it turns out that the sequence of function values f (1); f (p); f (p2); f (p3); : : : is linearly recursive, and hence we can associate a Tannakian symbol to the pair (f; p). Letting p vary over the set P of all prime numbers, we get a Pindexed C-valued Tannakian symbol attached to the multiplicative function f . For example, the Euler totient function has symbol fpg=f1g, the characteristic function of the square numbers has symbol f1; 1g=;, and the sum-of-divisors function has symbol f1; pg=;.
This assignment is injective on multiplicative functions, and for many classical classes of functions it stays injective even when U is reduced to a nite set of primes. Furthermore, Dirichlet convolution of functions correspond to addition of symbols, product of function corresponds to product of symbols (under a certain hypothesis), and norm operators on multiplicative functions correspond to certain Adams operations.
There are at least three interesting ways of associating a Tannakian symbol to a ( nite) graph. Firstly, given a graph X, we could de ne the Tannakian symbol of X to be A=;, where A is the spectrum of X, i.e. the multiset of eigenvalues of the adjacency matrix of X. With this de nition, taking the disjoint union of graphs would correspond to addition of Tannakian symbols, and taking tensor product of graphs would correspond to multiplication of Tannakian symbols. Graphs with the same spectrum are called isospectral, so the bers of this assignment would be classes of isospectral graphs.
Example 10. With this de nition, the Tannakian symbol of the complete graph on 4 vertices would be f 1; 1; 1; 3g=; It is conjectured that almost all graphs are determined by their spectra. However, there are many cases of non-isomorphic graphs with identical spectrum. For example, the number of simple graphs on 9 vertices is 274668 (see OEIS: Sequence A000088), while the number of such graphs isospectral to at least one other graph is 51039 (OEIS: Sequence A099883).
A second approach would be to de ne the Tannakian symbol of a graph X to be A=;, where A = f 1; 2; : : : ; mg is the nite multiset of complex numbers appearing in the expression
X (T ) = 1 2T ) (1 1T )(1 (1 mT ) where X (T ) is the Ihara zeta function of the graph X.
Example 11. With this alternative de nition, the Tannakian symbol of the
complete graph on 4 vertices would be
f 1; 1; 1; 1; 1; 2;
1+p7i ;
2
1+p7i ;
2
1+p7i ;
2
1
2
p7i ;
1
2
p7i ;
1
2
p7i =
g ;
In a recent paper [
As a third possibility, one can associate a certain polynomial (the \graph
polynomial") to any graph X. This polynomial de nes a scheme, called the graph
hypersurface of X, and we could associate Tannakian symbols to the graph X by
counting points of its graph hypersurface, like we did in the previous section for
an elliptic curve and the quartic Dwork family. We refer to Brown and Schnetz [
Let G be a nite group. Associated to G is its representation ring R(G), a lambda-ring generated by irreducible complex representations under the operations of direct sum, tensor product, and exterior powers. Any element g 2 G gives rise to a lambda-ring homomorphism from R(G) to TS(M ), where M is the monoid of complex roots of unity. Combining several such maps, one obtains a lambda-ring homomorphism E from R(G) into TSU (M ), where U is a subset of G. The most interesting choice of U , which we will use in the remainder of this section, is to pick one representative of each conjugacy class of G; this choice guarantees the injectivity of E.
Many interesting patterns and unsolved problems about representations can be reformulated in terms of the map E : R(G) ! TSU (M ). This is due to the fact that both the character table of G and the lambda-ring structure of R(G) can be recovered from the values of E.
Example 12. Taking G to be the Monster group, the character table is a 194
by 194 matrix, whose rank is 163. The number 163 also appears in the study
of imaginary quadratic number elds; it is in fact the largest possible integer D
such that the number eld Q(p D) has class number 1 (meaning that its ring
of integers is a unique factorization domain). The study of such number elds
goes back to Gauss and is the simplest instance of Gauss' famous class number
problem. As explained for example in the popular book of Mark Ronan [
An even more spectacular pattern connected with the Monster group is the Monstrous Moonshine Conjecture, formulated by Conway and Norton and proved by Richard Borcherds (Fields medal 1998). The starting point of this wonderful story was the observation that a certain number obtained from the character table (the dimension of the smallest nontrivial irreducible representation) is (almost) equal to the coe cient of the linear term in the Fourier expansion of Klein's j-function.
Let's now turn to a much simpler group, for which everything can be worked out by hand.
Example 13. One of the simplest non-trivial examples of a nite group is the symmetric group S3 of permutations on three objects. This group has 6 elements in total, partitioned into 3 conjugacy classes. Let e be the identity element, let t be any transposition (an element of order 2), and let r be one of the two \rotations" (an element of order 3). These three elements represent the three conjugacy classes of S3. The number of irreducible representations of a nite group is the same as the number of conjugacy classes, and in our example, the irreducible representations are: C+: The trivial representation, sending every permutation to 1.
C : The sign representation, sending a permutation to its sign.
C2: A two-dimensional representation, visualized as a matrix action of S3 on a triangle with vertices at the three cube roots of unity in the complex plane.
In this simple case, it is easy to compute the function E by hand. We get, for the three di erent choices of group element g: Case g = e : E(C+) = f1g=; Case g = t : E(C+) = f1g=; Case g = r : E(C+) = f1g=;
E(C ) = f1g=; E(C ) = f 1 =
g ; E(C ) = f1g=;
E(C2) = f1; 1g=; E(C2) = f1; 1 =
g ; E(C2) = f!; !2g=; Here ! is a primitive 3rd root of unity. Any element of R(G) can be written as a formal di erence V W , where V and W are representations built as direct sums of irreducible ones, and we may compute in R(G) by identifying such a formal di erence with an ordered triple of symbols (using the injective map E and applying the rules for computing with Tannakian symbols). For example, we get (suppressing E from the notation): C2 = f1; 1g=; ; f1; 1g=; ; f!; !2g=; and C+ C C2 = ;=; ; ;=; ; f1; 1g=f!; !2g . A similar computation shows that C2 C2 equals C+ C C2, and in general any tensor product of representations can be expressed as a direct sum of irreducibles, using only Tannakian symbols. 4.4
The arithmetic objects discussed so far in this paper were key players in many of
the greatest arithmetic discoveries of the 20th century. However, in 21st century
research, new classes of objects are becoming increasingly important, and these
objects come from homotopy theory and higher category theory. The most
beautiful application so far is probably the recent proof of the Tamagawa number
conjecture by Gaitsgory and Lurie [
Final remarks on arithmetic pattern-detection What constitutes an important discovery or a deep conjecture in arithmetic geometry? Looking at many examples in the literature, a few of which we have seen in the present survey, it is reasonable to say that deep arithmetic statements are often observations of patterns that can be expressed in terms of Tannakian symbols.
But is it conceivable that automated algorithms really could have detected some of these patterns? And is it reasonable to expect machines to discover new patterns, maybe even of comparable interest to the Weil conjectures, Monstrous Moonshine, or arithmetic mirror symmetry? Although we do not claim to know the answer to these questions, we would like to end by suggesting two necessary features that such algorithms would have to incorporate in order to have any chance of making such discoveries. 5.1
In traditional data analysis, data points are given by vectors of real numbers (or rather oating point approximations), visualized as points in n-dimensional Euclidean space Rn. Two data points are then considered to have similar features if they are close with respect to some metric on Rn derived from the standard metric (x; y) 7! jx yj on the set of real numbers.
Similarly, in a traditional neural network based on perceptrons or sigmoid neurons, the output of an individual neuron is a function of the size of an incoming real number, and \size" here refers to a measurement made using the standard metric on the real numbers.
In arithmetic geometry, many patterns and relations can be expressed in terms of a metric, but it is not enough to work with the standard metric on real numbers. Instead, the standard metric needs to be complemented by others, most importantly the p-adic metrics. For any prime number p, the p-adic metric on the set of rational numbers is de ned as follows. For two distinct rational numbers x and y, there is a unique integer k such that x y can be written on the form pk a=b, where a and b are positive integers coprime to p. The p-adic distance jx yjp between x and y is de ned to be p k. This de nition can be extended to irrational algebraic numbers, but unlike the standard metric, it is not de ned for transcendental numbers.
Unravelling the de nition, it is easy to see that as a special case, two integers x and y are congruent modulo p if and only if they are close in the sense that their p-adic distance is less than or equal to 1=2. Combining di erent primes and using the Chinese remainder theorem, any arithmetic pattern involving congruences can be expressed (and hence potentially discovered) using p-adic metrics2. 5.2
Another class of arithmetic patterns can be collected under the umbrella of symmetry. While the lambda-ring structure on Tannakian symbols in itself captures certain kinds of symmetry, there are also many other kinds, including Poincare duality (seen in the symbol of a projective scheme without singularities), various symmetries in Hodge diamonds, symmetries of modular forms, and the fact that the number of rows in a character table equals the number of columns.
A mathematical treatment of symmetry invariably involves group theory,
and in situations where the symmetry group is both nite and known, it is easy
to implement algorithms for detecting symmetry. However, in cases where the
symmetry group is in nite and/or unknown, the symmetry detection problem
2 It might also be interesting to build algorithms based on other kinds of metrics, like
the I-adic metric on a polynomial ring or a Grothendieck ring (where I is an ideal
of the ring), or the Granville-Soundararajan metric on multiplicative functions [