Philadelphia Area Number Theory Seminar

Tea and snacks beforehand, starting at 3pm.

Let K be a number field and let V be a "nice" p-adic representation of the absolute Galois group of K. The Iwasawa main conjectures attach arithmetic significance to special values of the p-adic L-function attached to V. Euler systems are a powerful tool for proving such conjectures; constructing Euler systems is difficult, though, and few examples exist. We will discuss applications of a new method for constructing Euler systems, first pioneered by Sangiovanni and Skinner. To explain the key ideas we focus on a simple example (based on joint work with Shang and Skinner): we reconstruct the two simplest Euler systems, the cyclotomic and elliptic units (corresponding to V=\Q_p(1), and K=\Q or K=an imaginary quadratic field, respectively). The Euler system classes are defined as extensions of Galois representations within the étale cohomology of the modular curve relative to a collection of points (cusps or CM points, respectively). We use Eisenstein series to construct special classes in cohomology; these Eisenstein series are naturally connected to the relevant p-adic L-functions in both cases. Afterwards, we discuss new applications of this method, including forthcoming work of the speaker to construct an Euler system for the “triple product” within the cohomology of the Siegel threefold.

Let K be a field with a nonarchimedean valuation, and let C be a curve over K defined by an equation of the form y^p = f(x), where p is any prime (which is allowed to be the residue characteristic of K).  The shape of a semistable model of such a curve can be determined from the cluster data of the roots of the polynomial f.  I will explain a way to encode such cluster data as a metric graph and, using this framework, provide a criterion for C to have a special geometric property called split degenerate reduction.  Meanwhile, this property is equivalent to C being uniformizable as a certain subset of the projective line modulo the action of a group of fractional linear transformations.  I will use this uniformization to demonstrate another perspective on the cluster data of a superelliptic curve with split degenerate reduction.

Isogeny graphs of supersingular elliptic curves have broad application, from the study and computation of modular forms to post-quantum cryptography. This is in part because the family of q-isogeny graphs of supersingular elliptic curves in characteristic p (with prime p varying, for a fixed prime q) is Ramanujan. One tool for studying a graph is its Ihara zeta function, defined as an Euler product over the primes of the graph. Defining the zeta function formally requires a graph in the sense of Serre and Bass, i.e. a directed graph equipped with a fixed-point free involution on the edge set. In general, isogeny graphs fail to be graphs in this sense. In this talk, I will discuss joint work with Lau, Orvis, Scullard, and Zobernig in which we introduce abstract isogeny graphs along with their zeta functions; these graphs capture the combinatorial structure of supersingular isogeny graphs (with level structure) . I will survey some of our results, including an analogue of Ihara’s determinant formula, showing in particular that the zeta function is rational. We use this formula and the Eichler-Shimura relation to give a formula relating the zeta function of a q-isogeny graph with level-H structure (for certain H, including B0(N) and B1(N)) to the Hasse-Weil zeta functions of two associated modular curves over the finite field Fq, generalizing results of Hashimoto, Sugiyama, and Lei-Muller.

Following Kneser, we survey the notion of neighboring lattices and how the study of the neighboring relation on lattices gives rise to modular forms. We proceed to study natural maps between spaces of modular forms induced by theta series, and relate it to classical questions in number theory.

Abstract: In Mazur’s celebrated Eisenstein ideal paper, he studies congruences between prime-level cusp forms and the unique weight-2 Eisenstein series of the same level. He shows that (if p is at least 5) such mod-p congruences exist if and only if the level N is congruent to 1 modulo p. In this talk, Jaclyn Lang considers Eisenstein–cuspidal congruences in weight 2 and level N2, still under the condition that N = 1 mod p. In this case, recent work with Pollack and Wake shows that the relevant level-N2 Hecke algebra is a free module over an appropriate inertia-at-N pseudodeformation ring. This structure turns out to be surprisingly powerful. One can recover Mazur’s existence theorem that there exists a mod-p Eisenstein–cuspidal congruence in weight 2 and prime level N when N = 1 mod p. It also allows one to deduce the relevant R=T theorem (for an appropriate pseudodeformation ring R) from the corresponding theorem in prime level, which is due to Wake and Wang–Erickson. Finally, one can recover the results of Merel and Lecouturier that characterize the rank of Mazur’s Hecke algebra in terms of the order of vanishing of a certain zeta element in cases when that rank is at most 3, and Lang explains some of the ideas that go into the aforementioned results. In addition to the joint work with Pollack and Wake, some of these results are joint with Palvannan and Müller.

Abstract: In his pioneering paper, Ribet uses Eisenstein congrunces to prove the converse to Herbrand's theorem. A key ingredient in his proof is the so-called Ribet's lemma of constructing nontrivial extensions of certain characters.  In my talk, I will go over Ribet's lemma and its various generalizations (due to Bellaiche and Urban). In particular, I will define symplectic extension graphs associated with symplectic representations, and study it by the use of Bruhat--Tits buildings associated with symplectic groups. We obtain a generalization of Urban's result and a symplectic analogue of the result of Bellaiche. 

Abstract: Researchers studying the inverse Galois problem realized 25 of the 26 sporadic finite simple groups as Galois groups over $\mathbb{Q}$ during 1984--1989. We complete this program by proving that the last remaining sporadic group, the Mathieu group $M_{23}$, occurs as a Galois group over $\mathbb{Q}$. In fact, we produce an explicit degree-$23$ polynomial with rational coefficients whose splitting field has Galois group $M_{23}$ over $\mathbb{Q}$. To accomplish this, we discover an unanticipated splitting of the Nielsen class associated to a non-rigid triple of conjugacy classes of $M_{23}$.  We compute the Belyi maps associated with this Nielsen class to construct an explicit regular Galois extension of $\mathbb{Q}(t)$ with Galois group $M_{23}$. Essential for our computation is the numerical Belyi map algorithm developed and implemented by Klug, Musty, Schiavone, Sijsling, and Voight, inspired by ideas of Hejhal and Stark. As a bonus, one specialization of the regular extension yields an $M_{22}$-extension of $\mathbb{Q}$. This is joint work with Blake Jackson, Kyu-Hwan Lee, Bjorn Poonen, Rachel Pries, and Shaowu Zhang.

Abstract: TBA

Abstract: TBA

Abstract: TBA