In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

# Past Number Theory Seminar

The talk will discuss the mean value theorem and Wooley's breakthrough with his "efficent congruencing" method.

I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

I describe joint work with Alastair Irving in which we improve a result of

D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to

contain $m$ primes. Here $m$ should be thought of as large but fixed, while $N$

tends to infinity.

The Harman sieve is the key to the improvement. The preprint will be

available on the Math ArXiv before the date of the talk.

Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

**Joint seminar with Logic.

I will discuss some recent results on the distribution of the real-analytic Eisenstein series on thin sets, such as a geodesic segment. These investigations are related to mean values of the Riemann zeta function, and have connections to quantum chaos.

The Cohen-Lenstra heuristics, postulated in the early 80s, conceptually explained numerous phenomena in the behaviour of ideal class groups of number fields that had puzzled mathematicians for decades, by proposing a probabilistic model: the probability that the class group of an imaginary quadratic field is isomorphic to a given group $A$ is inverse proportional to $\#\text{Aut}(A)$. This is a very natural model for random algebraic objects. But the probability weights for more general number fields, while agreeing well with experiments, look rather mysterious. I will explain how to recover the original heuristic in a very conceptual way by phrasing it in terms of Arakelov class groups instead. The main difficulty that one needs to overcome is that Arakelov class groups typically have infinitely many automorphisms. We build up a theory of commensurability of modules, of groups, and of rings, in order to remove this obstacle. This is joint work with Hendrik Lenstra.

Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Francesco Cellarosi.

I will review some classical problems in number theory concerning the statistical distribution of the primes, square-free numbers and values of the divisor function; for example, fluctuations in the number of primes in short intervals and in arithmetic progressions. I will then explain how analogues of these problems in the function field setting can be resolved by expressing them in terms of matrix integrals.