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Algebra and Number Theory Seminar - Fall 2021

3:30 pm EDT, Monday, September 20, 2021, 343 Kemeny, Masks Required

John Voight

Dartmouth College

Stickelberger's Discriminant Theorem for Algebras

Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over Z using techniques from linear algebra. Our proof, which only relies on elementary matrix identities, is new even in the classical case. This is joint work with Asher Auel and Owen Biesel.

3:30 pm EDT, Monday, October 11, 2021, 343 Kemeny, Masks Required

Avi Kulkarni

Dartmouth College

Intersections of symmetric determinantal varieties, theta characteristics, and an application to arithmetic.

A symmetric determinantal hypersurface (symmetroid) is the zero locus of the determinant of a symmetric matrix of linear forms. In this talk, we study intersections of symmetroids. First, we classify when an intersection of symmetroids contains an accidental'' singularity. We then use this result to study theta characteristics of complete intersection canonical curves. Finally, we discuss the inverse Galois problem for del Pezzo surfaces of degree one. Joint work with Sameera Vemulapalli.

3:30 pm EDT, Monday, October 18, 2021, 343 Kemeny, Masks Required

Asher Auel

Dartmouth College

Families of elliptic curves of positive rank

In this talk, I'll describe a construction of new families of elliptic curves of positive rank coming from the theory of genus one curves splitting Brauer classes. This is joint work with Danny Krashen.

3:30 pm EDT, Monday, October 25, 2021, 343 Kemeny, Masks Required

Jack Petok

Dartmouth College

Composite level images of Galois representations

In the past few years, much progress had been made towards the classification of the images of prime-level Galois representations arising from elliptic curves defined over the rational numbers. A fruitful approach to this problem is to directly compute rational points on the modular curves parametrizing elliptic curves with prescribed image of Galois. Using what we know today about rational points on prime-level modular curves, we compute rational points on many composite-level modular curves. As a consequence, we produce a classfication of the groups of level (m_1, m_2) which occur for infinitely many rational elliptic curves, where (m_1, m_2) are powers of distinct primes less than 17. This is joint work with Catalina Camacho, Wanlin Li, Jackson Morrow, and David Zureick-Brown.

3:30 pm EDT, Monday, November 8, 2021, 343 Kemeny, Masks Required

Grant Molnar

Dartmouth College

Counting 7-isogenies

In this talk, we discuss asymptotics for the number of elliptic curves of height up to X which are equipped with a 7-isogeny. Even solving this problem for curves with rational coefficients up to Q^alg-isomorphism requires some delicacy, because they are parameterized by polynomials that share a common factor. This problem is magnified when working with elliptic curves up to Q-isomorphism, which requires summing over quadratic twists. We report partial progress and avenues for future investigation.

3:30 pm EDT, Monday, November 15, 2021, 343 Kemeny, Masks Required

Juanita Duque-Rosero

Dartmouth College

Enumerating triangular modular curves of low genus

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves arise from Belyi maps with monodromy PGL_2(F_q) or PSL_2(F_q). In this talk, we will give an idea on the construction of these curves and we will present a computational approach to enumerate all triangular modular curves of genus 0, 1, and 2. This is joint work with John Voight.