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\centerline{\bf What is Arithmetic Statistics?}

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Number Theory has its share of conjecture and heuristics that thrive on---if not depend on---the accumulation of aggregates of instances, aggregates of numerical data.

\vskip20pt To see that {\it numerical data related to numbers themselves} is also at the very heart of the pleasure of number theory, and is a major reason for the very {\it theory} itself, consider this letter of Gauss to one of his students (the {\it italics} are ours): \vskip20pt \begin{quote} Even before I had begun my more detailed investigations into higher arithmetic, one of my first projects was to turn my attention to the decreasing frequency of primes, to which end {\it I counted the primes in several chiliads} and recorded the results on the attached white pages. I soon recognized that behind all of its fluctuations, this frequency is on the average inversely proportional to the logarithm, so that the number of primes below a given bound n is approximately equal to $$\int dn/ \log(n),$$ where the logarithm is understood to be hyperbolic. Later on, when I became acquainted with the list in Vega?s tables (1796) going up to 400031, I extended my computation further, confirming that estimate. {\it In 1811, the appearance of Chernau?s cribrum gave me much pleasure and I have frequently (since I lack the patience for a continuous count) spent an idle quarter of an hour to count another chiliad here and there\dots}\end{quote} \vskip20pt

• %One aspect of number theory (and probably any branch of mathematics) is the way that {\it computation} (i.e., close observation of a significant mass of the phenomena under study) weaves in and out with {\it theory}.
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  Often, in modern number theory, to actually sample a sufficient quantity of

data that might allow you to guess even approximate qualitative behavior of the issue you are studying, you may have to go out to very high numbers. For example, there are basic questions about elliptic curves (e.g., what is the frequency of those possessing two independent rational points of infinite order) where if you only look at curves of conductor < 10^8, you might be tempted to make guesses that are not only wrong, but qualitatively wrong.

• \vskip20pt Our program {\it Arithmetic Statistics} then stands for those

aspects of number theory---be it theory or computation---that connect closely with concrete (important) {\it numerical data related to numbers themselves}. \vskip15pt Many people in our program are engaged in the theoretical side of our subject, and many in the computational side. Much theoretical" modern number theory bears on, and sometimes has vital need of large scale computing projects and large data-bases. And both the computational and theoretical facets connect to some of the famous heuristics in our subject: {\it Cohen-Lenstra} heuristics (average expected size of various finite abelian groups that appear in our subject; and {\it Random matrix} heuristics.

% Fundamental issues include the statistics regarding zeroes of L-functions; arithmetic questions about elliptic curves---average size of Mordell-Weil ranks, and Selmer ranks; upper bounds on trigonometric sums; etc. And we have people in our program who are at the center of this activity.

\vskip10pt The computational and theoretical facets of our subject form one interlocking unity. %Our program bridges these facets: people working in each of them will have ample opportunity to interact with people working in the other mode.

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• Here are a few examples of recent work that have been themes of our program:

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  • \begin{itemize} \item Manjul Bhargava together with his students and

co-authors have been developing extremely precise methods for counting appropriate orbits of certain arithmetic groups acting on integral points on certain lattices. This approach follows and significantly refines the classical Methods in the Geometry of Numbers (as had pursued by Gauss, Minkowski, Siegel, and others). A major application of this work of Bhargava and co-authors is to establish counts of important ingredients of the arithmetic of elliptic curves. Among their applications is the result of Bhargava-Shankar that the average rank of the Mordell-Weil group of elliptic curves over {\bf Q}---when they are ordered in any of the standard ways---is <1.5.

• This result is related to their study of the average {\it size} of the

2-Selmer rank of elliptic curves (again over {\bf Q} ---and when they are ordered in any of the standard ways). They show that the average size is {\it three}{\footnote{Of course no 2-Selmer group can have such a size: these 2-Selmer groups are then all either above or below average.}} For any prime number p the {\it reduced p-Selmer rank of an elliptic curve over a number field}{\footnote{ This is the dimension of the so-called p-Selmer group minus the rank of rational p-torsion of the elliptic curve over the number field.}} has this important property: it is finite (!), computable (at least in principle), and is an upper bound for the rank of the Mordell-Weil group of the elliptic curve over the number field. If the Shafarevich-Tate conjecture holds, then for all but finitely many primes p, the reduced p-Selmer rank would be equal to that Mordell-Weil rank. So it is natural, as in the results of Bhargava and co-authors alluded to above, to expect that the statistics of p-Selmer ranks (e.g., even when restricted to p=2) contribute to our understanding of Mordell-Weil ranks. In the course of our program we have been learning the most recent advances in this direction (3-Selmer, 5-Selmer).

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  \item The heuristics predicting average sizes" of quite a few important

arithmetic objects have also been the focus of our program. We are fortunate to have both Henri Cohen and Hendrik Lenstra among us. They are the co-originators of the {\it Cohen-Lenstra heuristics} that guides conjectures regarding average sizes of ideal class groups and other important invariants in number theory. The latest development in the formidable toolbox of heuristics is due to Bjorn Poonen and Eric Rains and has a somewhat different slant; it gives one precise guesses for the probabilities of reduced p-Selmer ranks for elliptic curves over a given number field (when these curves are ordered in the usual way). This too has been one of the focusses of our program. A few years ago, Peter Swinnerton-Dyer, extending earlier results of Heath-Brown, studied the probabilities of reduced 2-Selmer ranks of families of elliptic curves that are quadratic twists of some very specific types of elliptic curves over {\bf Q}. One grand (and enticing) feature of Swinnerton-Dyer's study is that the probabilities arise as if they were the product of a specific Markov process; another curious feature---a drawback, perhaps---is that the statistics are dependent upon ordering the elliptic curves in the twist family {\it not in the standard way} but in terms of the number of primes dividing the discriminant. All the issues that are brought up by this work are focusses of current research in our program. Specifically, Dan Kane's work in the program is towards relating such Swinnerton-Dyer statistics dependent upon different orderings of the collection of elliptic curves being sampled, while Karl Rubin, Zev Klagsbrun, and Barry Mazur are developing an approach (which has a Markov Process feel") to unconditionally prove the expected statistics for reduced 2-Selmer ranks over an arbitrary number field for all quadratic twists families of many elliptic curves (the elliptic curves in any of these families are ordered in a certain not entirely unnatural, but again non-standard, way). \vskip10pt \item Dirichlet L-functions are the simplest generalizations of the Riemann zeta-function. They were invented by Dirichlet and have been used to prove an asymptotic formula for the number of primes up to a quantity X in a given arithmetic progression modulo q. Like the Riemann zeta-function each Dirichlet L-function can be expressed as Dirichlet series (the Riemann zeta-function has Dirichlet series coefficients 1,1,1,\dots and the first Dirichlet L-function has coefficients that repeat mod 3: 1,-1,0,1,-1,0,\dots), has a functional equation and Euler product, and is conjectured to have its zeros on the 1/2-line; the latter assertion is sometimes called the Generalized Riemann Hypothesis. It can be proven that each individual Dirichlet L-function has at least 40\% of its zeros on the 1/2-line. Conrey, Iwaniec, and Soundararajan have now shown that when all of the zeros of these Dirichlet L-functions are taken together at least 55\% of these zeros are on the 1/2-line. To be specific, take a large number Q and consider all of the L-functions associated with a primitive character modulo q where $q\le Q$. Now consider all of the zeros of all of these L-functions which are located in the rectangle of complex numbers with real parts between 0 and 1 and imaginary parts between 0 and \log Q. CIS can prove that at least 55\% of the zeros in this rectangle have real parts equal to 1/2.

The technique used by CIS is something they call the asymptotic large sieve. This is a technique which can be used to give an asymptotic formula for a quantity that would have previously been estimated by the {\it large sieve inequality.} The latter has been a staple of number theorists for more than 4 decades now. One spectacular application of the large sieve inequality is to prove the Bombieri-Vinogradov theorem which asserts that when counting primes up to X in arithmetic progressions with moduli up to Q then the error terms behave, on average, as well as could be expected, that is, as well as could be proved assuming the Generalized Riemann Hypothesis. Not surprisingly, the Bombieri-Vinogradov theorem is a much celebrated result. Indeed, Enrico Bombieri won the Fields medal in 1974 for this work. A few years ago Goldston, Pintz, and Yildirim used the BV theorem to prove their much celebrated theorem that the smallest gaps between consecutive prime numbers are an order of magnitude smaller than the average gaps. Now, with their asymptotic version of the large sieve, CIS have studied zeros on the 1/2-line, not only of Dirichlet L-functions, but of other families as well: twists of a fixed L-function of degree 2 by Dirichlet characters (at least 36\% of their zeros are on the 1/2-line) and twists of degree 3 L-functions (at least one-half of one percent of their zeros are on the 1/2-line). In addition, CIS have been able to confirm a prediction from Random Matrix Theory about the sixth moment of Dirichlet L-functions at the point 1/2, averaged over characters with moduli up to Q. They prove a formula which includes all of the main terms and has an error term which is a power of Q smaller than the main terms The main terms are expressed in terms of simple factors multiplied by a ninth degree polynomial in \log Q. The leading coefficient of the polynomial is 42 and the lower terms are given explicitly in terms of complicated arithemetic and geometric factors. The theorem {\bf exactly} matches the predictions arising from Random Matrix Theory, and provides excellent confirmation of the RMT models for L-functions.

• Besides the three researchers just discussed, we are very fortunate to have

as participants some of the pioneers who have developed probabilistic models for L-functions, including Roger Heath-Brown, David Farmer, Jon Keating, Francesco Mezzadri, Michael Rubinstein, Peter Sarnak, and Nina Snaith.

• \item Several of our rsearchers are examining statistics for

curves over finite fields. The zeros of the zeta function are the inverses of the eigenvalues of the Frobenius endomorphism. The work of Katz and Sarnak indicates that when g is fixed and q tends to infinity, the normalized zeros are distributed like the eigenvalues of matrices in a group of random matrices determined by the monodromy group of the moduli space of C. But the related question of studying statistics as q remains fixed and the genus g grows to infinity is still largely unknown, though recent progress has been made in computing the distribution of the trace of the Frobenius endomorphism for various families by Kurberg-Rudnick, Bucur-David-Feigon-Lal\'in and Bucur-Kedlaya.

The broader question of computing the global distribution of the zeros in the g limit remains. This is a non-trivial modeling job, since the global obstruction imposes an infinite, but discrete, set of conditions that the matrix model should satisfy. Such a model needs to exhibit both discrete and continuous features in order to capture the global phenomenon. Bucur and Feigon, together with their collaborators, David and Lal\'in will be working in this direction while at MSRI.

• \item Computation and experimentation have been playing a large role in our

program. For example, postdocs Jonathan Bober and Ghaith Hiary have been implementing Hiary's world's fastest algorithms for the Riemann zeta function, computing zeros of \zeta(s) with \Im{s} near 10^{36}. Andrew Booker and Michael Rubinstein are carrying out computations of zeros of more general L-functions, gathering extensive numerical evidence in favour of the Riemann Hypothesis. William Stein has been tabulating elliptic curves over Q(\sqrt(5)), and verifying the Birch and Swinnerton-Dyer conjecture. Nathan Ryan, Nils Skoruppa, Fredrik Str\"omberg, and Gonzalo Tornaria (NNFG) have been studying methods for computing with Siegel modular and paramodular forms which have degree 4 and 5 L-functions associated to them. Duc Khiem Huynh is attempting to develop probabilistic models for these L-functions with the goal of testing the predictions using data provided by (NNFG). Nils Skoruppa works on a new algorithm for computing modular forms of half integral weight directly from the periods of the associated modular forms of integral weight. This will make it possible to tabulate half integral modular forms of very high level without the need of computing complete (and then very high dimensional) spaces as is required by the currently known algorithms. David Farmer, Stefan Lemurell, and Sally Koutsoliotas are developing methods for finding Maass forms for higher rank groups and testing conjectures regarding their Fourier coefficients and associated L-functions. Cremona.... Hanke...

• In the study of elliptic curves over totally real number fields like

Q(\sqrt 5) (recent work of William Stein) one is naturally led to Hilbert modular forms. Work of Shimura and recently of Ikeda in Japan indicates that there is a similar connection between modular forms of half integral weight and modular forms of integral weight over number fields as it is well-known for Q. However, as it is known from the theory over Q it has several advantages to replace in such a theory the modular froms of half integral weight by Jacobi forms. The Fourier coefficients of these Jacobi forms correspond (in the theory over Q) to the central value in the critical strip of the twisted L-series of the associated Hilbert modular forms or elliptic curve over Q. Skoruppa and his student Hatice Boylan are preparing a longer article to set up such a theory over arbitrary number fields too based on results of Boylan's thesis. In particular, they want, in joint work with Fredrik Str\"omberg, to compute sufficiently many examples of Jacobi forms over Q(\sqrt 5) which should complement the computations of William Stein on elliptic curves over $Q(\sqrt 5)$.

• \item \dots

\end{itemize}

Learning seminars, whereby our participants meet weekly to teach each other and discuss material relevant to our research, forms an important part of our program. The Bhargava-Shankar group has been meeting to learn material related to the work of Bhargava and Shanlar on ranks of elliptic curves. The explicit formula group is studying the problem of ranks from an analytic perspective. The low lying zeros seminar has been looking at papers related to the distribution of zeros in families of L-functions. Quadratic twists of elliptic curves meets to discuss the problem of ranks of elliptic curves in fmailies of quadratic twists. Earlier, a group was meeting to study the Cohen-Lenstra heuristics and its extension to Tate-Shafarevich groups by Christophe Delaunay. Lastly, a few researchers are holding a seminar to study paramodular forms.

The first workshop to take place as part of the Arithmetic Statistics program was the 2-day Connections for Women event. This targeted female mathematicians in fields related to the program, but we were pleased to see that all aspects of the workshop were well-attended by the program's participants, which lead to a very even mix of male and female researchers. The Connections for Women workshop was a very agreeable mixture of excellent talks, a buzz of mathematical discussion and a chance to meet new people; every math workshop should be like this! The audience enjoyed 6 superb talks by leading women in the area, ranging from the number theory involved in cryptography to several of the questions of counting (ranks, points on curves, number fields) that are themes of the rest of the program.

The discussion session on pursuing a career in mathematics saw senior mathematicians giving advice on how to apply for first jobs and postdoctoral positions, some anecdotes about how dual-career couples have found posts in the same institution, and strategies for departments keen to increase the number of women in their faculty. With participants covering the spectrum from undergraduates to those with a long career behind them, the discussion was lively and productive.

These two days then lead into the main Introductory workshop for the Arithmetic Statistics program, which most of the Connections participants stayed on to enjoy.

Three other workshops form a part of our program. Our introductory workshop was held from January 31-February 4 and featured talks to help define the direction of our program. Talks were given, in order of appearance, by: Henri Cohen, Karl Rubin, Manjul Bhargava, Michael Rubinstein, Nina Snaith, Melanie Wood, Brian Conrey, Andrew Sutherland, Jordan Ellenberg, David Farmer, John Voight, Henryk Iwaniec, Akshay Venkatesh, John Cremona, Bjorn Poonen, William Stein, Kannan Soundararajan, Chantal David, and Frank Thorne.

Several of the partipants in our program are also involved in a large scale NSF funded collborative Focused Research Group project to develop methods for computing with L-functions and associated automorphic forms, as well as verify many of the important conjectures in this area. In order to help diffuse the large mount of data being generated by the project, an archive with a user friendly front end for browsing and searching the data is being developed, and a workshop involving 15 participants was held at the MSRI, Feb 21-25, to continue developing the archive.

The last workshop for our program will be held April 11-15 on the theme of `Arithmetic Statistics' and it will highlight some of the work being carried out at the MSRI during our program. In order to give participants more opportunity to interact and collaborate fewer talks are being planned.

{\bf Profile: Melanie Wood and Manjul Bhargava}

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