Chebyshev famously observed empirically that more often than not, there are more primes of the form 3 mod 4 up to x than primes of the form 1 mod 4. This was confirmed theoretically much later by Rubinstein and Sarnak in a logarithmic density sense. Our understanding of this is conditional on the generalized Riemann Hypothesis as well as Linear Independence of the zeros of L-functions.
We investigate similar questions for sums of two squares in arithmetic progressions. We find a significantly stronger bias than in primes, which happens for almost all integers in a natural density sense. Because the bias is more pronounced, we do not need to assume Linear Independence of zeros, only a Chowla-type Conjecture on non-vanishing of L-functions at 1/2.
We'll aim to be self-contained and define all the notions mentioned above during the talk. We shall review the origin of the bias in the case of primes and the work of Rubinstein and Sarnak. We'll explain the main ideas behind the proof of the bias in the sums-of-squares setting.
In 2004, Montgomery and Soundararajan showed (conditionally) that the distribution of the number of primes in appropriately sized intervals is approximately Gaussian and has a somewhat smaller variance than you might expect from modeling the primes as a purely random sequence. Their work depends on evaluating sums of certain arithmetic constants that generalize the twin prime constant, known as singular series. In particular, these sums exhibit square-root cancellation in each term if they have an even number of terms, but if they have an odd number of terms, there should be slightly more than square-root cancellation. I will discuss sums of singular series with an odd number of terms, including tighter bounds for small cases and the function field analog. I will also explain how this problem is connected to a simple problem about adding fractions.
Conrey, Farmer and Zirnbauer formulated the ratios conjectures, which give asymptotic formulas for the ratios of products of shifted L-functions from some family. They have many corollaries to other problems in arithmetic statistics, such as the computation of various moments or the distribution of zeros in a family of L-functions.
During the talk, we will show how to use multiple Dirichlet series to prove the conjectures in the family of real Dirichlet L-functions for some range of the shifts. The talk will be accessible even to those with little background in analytic number theory.
We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length $y$ around $x$, where $y\ll(\log x)^2$. In particular, we conjecture that the maximum grows surprisingly slowly as $y$ranges from $\log x$ to $(\log x)^2$. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.
We consider a crystalline universal deformation ring R of an n-dimensional, mod p Galois representation whose semisimplification is the direct sum of two non-isomorphic absolutely irreducible representations. Under some hypotheses, we obtain that R is a discrete valuation ring. The method examines the ideal of reducibility of R, which is used to construct extensions of representations in a Selmer group with specified dimension. This can be used to deduce modularity of representations.
An integral ring R is a ring additively isomorphic to Z^n . The subring zeta function is an important tool in studying subring growth in these rings. One can compute these zeta functions using p-adic integration due to a result of Grunewald, Segal and Smith. I shall talk about computing these zeta functions for Z[t]/(t^n) for small n and describe some results on subring growth and ideal growth for integral rings. This includes joint work with Ramin Takloo-Bighash and Gautam Chinta.
The recent proofs of the Tate conjecture for K3 surfaces over finite fields start by lifting the surface to characteristic 0. Serre showed in the sixties that not every variety can be lifted, but the question whether every motive lifts to characteristic 0 is open. We give a negative answer to a geometric version of this question, by constructing a smooth projective variety that cannot be dominated by a smooth projective variety that lifts to characteristic 0.
In this talk, I will discuss how recent developments in arithmetic geometry (for example pertaining to perfectoid spaces) led to significant new discoveries in commutative algebra and algebraic geometry in mixed characteristic.
Let k be a perfect field of characteristic p>0 and let X be a proper scheme over W(k) with semistable reduction. I shall define a log-Chow group for the special fibre X_k and give an interpretation in terms of logarithmic Milnor K-theory. Then, by gluing a logarithmic variant of the Suslin-Voevodsky motivic complex to a log-syntomic complex along the logarithmic Hyodo-Kato Hodge-Witt sheaf, I will prove that an element of the r-th log-Chow group of X_k formally lifts to the continuous log-Chow group of X if and only if it is “Hodge” (i.e. its log-crystalline Chern class lands in the r-th step of the Hodge filtration of the generic fibre of X under the Hyodo-Kato isomorphism). This simultaneously generalises a result of Yamashita (which is the case r=1), and of Bloch-Esnault-Kerz (which is the case of good reduction). This is joint work with Andreas Langer.