The equidistribution theorem for rational points on expanding horospheres with fixed denominator in the space of d-dimensional Euclidean lattices has been derived in the work by M. Einsiedler, S. Mozes, N. Shah and U. Shapira. The proof of their theorem requires ergodic theoretic tools, including Ratner's measure classification theorem. In this talk I will present an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums. In the case of d=3, unlike the ergodic-theoretic approach, this provides an explicit estimate on the rate of convergence. This is a joint work with Jens Marklof.

# Past Number Theory Seminar

In 2000, Vojta solved the n-squares problem under the Bombieri-Lang conjecture, by explicitly finding all the curves of genus 0 or 1 on the surfaces related to this problem. The fundamental notion used by him is $\omega$-integrality of curves.

In this talk, I will show a generalization of Vojta's method to find all curves of low genus in some surfaces, with arithmetic applications.

I will also explain how to use $\omega$-integrality to obtain a bound of the height of a non-constant morphism from a curve to $\mathbb{P}^2$ in terms of the number of intersections (without multiplicities) of its image with a divisor of a particular kind.

This proves some new special cases of Vojta's conjecture for function fields.

I will revisit the theory of Hodge-Tate local systems in the light of the p-adic Simpson correspondence. This is a joint work with Michel Gros.

Abstract: We will recall some analogies between structures arising from three-manifold topology and rings of integers in number fields. This can be used to define a Chern-Simons functional on spaces of Galois representations. Some sample computations and elementary applications will be shown.

Let $F$ be a binary form of degree $d \geq 3$ with integer coefficients and non-zero discriminant. In this talk we give an asymptotic formula for the quantity $R_F(Z)$, the number of integers in the interval $[-Z,Z]$ representable by the binary form $F$.

This is joint work with C.L. Stewart.

Let $a_1, \cdots, a_N$ be the sequence of y-smooth numbers up to x (i.e. composed only of primes up to y). When y is a small power of x, what can one say about the size of the gaps $a_{j+1}-a_j$? In particular, what about

$$\sum_1^N (a_{j+1}-a_j)^2?$$

When considering $E_k$ numbers (products of exactly $k$ primes), it is natural to ask, how they are distributed in short intervals. One can show much stronger results when one restricts to almost all intervals. In this context, we seek the smallest value of c such that the intervals $[x,x+(\log x)^c]$ contain an $E_k$ number almost always. Harman showed that $c=7+\varepsilon$ is admissible for $E_2$ numbers, and this was the best known result also for $E_k$ numbers with $k>2$.

We show that for $E_3$ numbers one can take $c=1+\varepsilon$, which is optimal up to $\varepsilon$. We also obtain the value $c=3.51$ for $E_2$ numbers. The proof uses pointwise, large values and mean value results for Dirichlet polynomials as well as sieve methods.

Let $L/K$ be an extension of number fields and let $J_L$ and $J_K$ be the associated groups of ideles. Using the diagonal embedding, we view $L^*$ and $K^*$ as subgroups of $J_L$ and $J_K$ respectively. The norm map $N: J_L\to J_K$ restricts to the usual field norm $N: L^*\to K^*$ on $L^*$. Thus, if an element of $K^*$ is a norm from $L^*$, then it is a norm from $J_L$. We say that the Hasse norm principle holds for $L/K$ if the converse holds, i.e. if every element of $K^*$ which is a norm from $J_L$ is in fact a norm from $L^*$.

The original Hasse norm theorem states that the Hasse norm principle holds for cyclic extensions. Biquadratic extensions give the smallest examples for which the Hasse norm principle can fail. One might ask, what proportion of biquadratic extensions of $K$ fail the Hasse norm principle? More generally, for an abelian group $G$, what proportion of extensions of $K$ with Galois group $G$ fail the Hasse norm principle? I will describe the finite abelian groups for which this proportion is positive. This involves counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which is achieved using tools from harmonic analysis.

This is joint work with Christopher Frei and Daniel Loughran.

The sub-convexity barrier traditionally prevents one from applying the Hardy-Littlewood (circle) method to Diophantine problems in which the number of variables is smaller than twice the inherent total degree. Thus, for a homogeneous polynomial in a number of variables bounded above by twice its degree, useful estimates for the associated exponential sum can be expected to be no better than the square-root of the associated reservoir of variables. In consequence, the error term in any application of the circle method to such a problem cannot be expected to be smaller than the anticipated main term, and one fails to deliver an asymptotic formula. There are perishingly few examples in which this sub-convexity barrier has been circumvented, and even fewer having associated degree exceeding two. In this talk we review old and more recent progress, and exhibit a new class of examples of Diophantine problems associated with, though definitely not, of translation-invariant type.

For $k \geq 3$ we give new values of $\rho_k$ such that

$$ \| \alpha p^k + \beta \| < p^{-\rho_k} $$

has infinitely many solutions in primes whenever $\alpha$ is irrational and $\beta$ is real. The mean

value results of Bourgain, Demeter, and Guth are useful for $k \geq 6$; for all $k$, the results also

depend on bounding the number of solutions of a congruence of the form

$$ \left\| \frac{sy^k}{q} \right\| < \frac{1}{Z} \ \ (1 \leq y \leq Y < q) $$

where $q$ is a given large natural number.