arXiv subjects
Publications and source records attributed to Javier Fresán.
We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the geometric properties of generalized jacobians of curves and on general equidistribution theorems for character sums over finite fields.
We construct explicit families of graphs whose eigenvalues are asymptotically distributed according to Wigner's semicircle law; in other words, that are spectrally indistinguishable from random graphs. However, in other respects they are strikingly dissimilar from random graphs; for example, they are $K_{2,3}$-free graphs with almost the maximum possible edge density.
We study the arithmetic Fourier transforms of trace functions on general connected commutative algebraic groups. To do so, we first prove a generic vanishing theorem for twists of perverse sheaves by characters, and using this tool, we construct a tannakian category with convolution as tensor operation. Using Deligne's Riemann Hypothesis, we show how this leads to a general equidistribution theorem for the discrete Fourier transforms of trace functions of perverse sheaves, generalizing the work of Katz in the case of the multiplicative group. We then give some concrete examples of applications of these results and raise a number of questions.
We prove some old and new convergence statements for fixed-points statistics using tensor envelope categories, such as the Deligne--Knop category of representations of the "symmetric group" $S_t$ for an indeterminate~$t$. We also discuss some arithmetic speculations related to Chebotarev's density theorem.
Classical polylogarithms give rise to a variation of mixed Hodge-Tate structures on the punctured projective line $S=\mathbb{P}^1\setminus \{0, 1, \infty\}$, which is an extension of the symmetric power of the Kummer variation by a trivial variation. By results of Beilinson-Deligne, Huber-Wildeshaus, and Ayoub, this polylogarithm variation has a lift to the category of mixed Tate motives over $S$, whose existence is proved by computing the corresponding space of extensions in both the motivic and the Hodge settings. In this paper, we construct the polylogarithm motive as an explicit relative cohomology motive, namely that of the complement of the hypersurface $\{1-zt_1\cdots t_n=0\}$ in affine space $\mathbb{A}^n_S$ relative to the union of the hyperplanes $\{t_i=0\}$ and $\{t_i=1\}$.
We answer in the negative Siegel's problem for $G$-functions, as formulated by Fischler and Rivoal. Roughly, we prove that there are $G$-functions that cannot be written as polynomial expressions in algebraic pullbacks of hypergeometric functions; our examples satisfy differential equations of order two, which is the smallest possible. In fact, we construct infinitely many non-equivalent rank-two local systems of geometric origin which are not algebraic pullbacks of hypergeometric local systems, thereby providing further counterexamples to Dwork's conjecture and answering a question by Krammer. The main ingredients of the proof are a Lie algebra version of Goursat's lemma, the monodromy computations of hypergeometric local systems due to Beukers and Heckman, as well as results on invariant trace fields of Fuchsian groups.
We report new examples of Sidon sets in abelian groups arising from generalized jacobians of curves, and discuss some of their properties with respect to size and structure.
We prove the existence of quadratic relations between periods of meromorphic flat bundles on complex manifolds with poles along a divisor with normal crossings under the assumption of "goodness". In dimension one, for which goodness is always satisfied, we provide methods to compute the various pairings involved. In an appendix, we give details on the classical results needed for the proofs. V3: Revised version, various proofs simplified in Section 3, exposition improved.
This survey article is the outgrowth of two talks given at the Journées X-UPS "Périodes et transcendance" at École polytechnique. Periods are complex numbers whose real and imaginary parts can be written as integrals of rational functions over domains defined by polynomial inequalities, everything with rational coefficients. According to a conjecture by Kontsevich-Zagier, every algebraic relation among these numbers should follow from the obvious rules of calculus: additivity, change of variables, Stokes's formula. I first explain the definition of periods and some ensuing elementary properties, by illustrating them with a host of examples. Then I gently move to the interpretation of these numbers as entries of the integration pairing between algebraic de Rham cohomology and singular homology of algebraic varieties defined over $\mathbf{Q}$, the point of view which is at the origin of all recent breakthroughs in their study.
We construct motives over the rational numbers associated with symmetric power moments of Kloosterman sums, and prove that their L-functions extend meromorphically to the complex plane and satisfy a functional equation conjectured by Broadhurst and Roberts. Although the motives in question turn out to be "classical", we compute their Hodge numbers by means of the irregular Hodge filtration on their realizations as exponential mixed Hodge structures. We show that all Hodge numbers are either zero or one, which implies potential automorphy thanks to recent results of Patrikis and Taylor.
We answer in the negative Siegel's question whether all E-functions are polynomial expressions in hypergeometric E-functions. Namely, we show that if an irreducible differential operator of order three annihilates an E-function in the hypergeometric class, then the singularities of its Fourier transform are constrained to satisfy a symmetry property that generically does not hold. The proof relies on André's theory of E-operators and Katz's computation of the Galois group of hypergeometric differential equations.
Motivated by the computation of certain Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between the Bessel moments \[ \int_0^\infty I_0(t)^i K_0(t)^{k-i}t^{2j-1}\,\mathrm{d}t \qquad (i, j=1, \ldots, \lfloor (k-1)/2\rfloor), \] where $k \geq 1$ is a fixed integer and $I_0$ and $K_0$ denote the modified Bessel functions. In this paper, we interpret these integrals and variants thereof as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. Building on the general framework developed in arXiv:2005.11525, this enables us to prove quadratic relations of the form suggested by Broadhurst and Roberts, which conjecturally comprise all algebraic relations between these numbers. We also make Deligne's conjecture explicit, thus explaining many evaluations of critical values of $L$-functions of symmetric power moments of Kloosterman sums in terms of determinants of Bessel moments.
Many exponential sums over finite fields, including Gauss sums and Kloosterman sums, arise as the Fourier transform with respect to a character of the trace function of an $\ell$-adic sheaf on a commutative algebraic group. We study the equidistribution of these sums when the sheaf is fixed but the character varies over larger and larger extensions of the finite field. For the additive group, monodromy governs equidistribution by a theorem of Deligne. A few years ago, Katz solved the multiplicative variant of the question in a work where Tannakian ideas play an essential role.
We introduce the notion of algebraic cogroup over a subfield $k$ of the complex numbers, and use it to prove that every Nori motive over $k$ is isomorphic to a quotient of a motive of the form $H^n(X, Y)(i)$.
At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are products of values of the gamma function at rational arguments, with exponents determined by the Hodge decomposition. We prove an alternating variant of this conjecture for smooth projective varieties acted upon by an automorphism of finite order, thus improving previous results of Maillot and Rössler. The proof relies on a product formula for periods of regular singular connections due to Saito and Terasoma.
We investigate from a mathematical perspective how Feynman amplitudes appear in the low-energy limit of string amplitudes. In this paper, we prove the convergence of the integrands. We derive this from results describing the asymptotic behavior of the height pairing between degree-zero divisors, as a family of Riemann surfaces degenerates. These are obtained by means of the nilpotent orbit theorem in Hodge theory.