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september 28 |
organizational meeting (no zoom) |
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october 5 |
aaron pollack (uc san diego)
quaternionic modular forms (qmfs) on the split exceptional group g_2 are a special class of automorphic functions on this group, whose origin goes back to work of gross-wallach and gan-gross-savin. while the group g_2 does not possess any holomorphic modular forms, the quaternionic modular forms seem to be able to be a good substitute. in particular, qmfs on g_2 possess a semi-classical fourier expansion and fourier coefficients, just like holomorphic modular forms on shimura varieties. i will explain the proof that the cuspidal qmfs of even weight at least 6 admit an arithmetic structure: there is a basis of the space of all such cusp forms, for which every fourier coefficient of every element of this basis lies in the cyclotomic extension of q. |
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october 19 |
christian klevdal (uc san diego)
in this talk, we study a tannakian category of admissible pairs, which arise naturally when one is comparing etale and de rham cohomology of p-adic formal schemes. admissible pairs are parameterized by local shimura varieties and their non-minuscule generalizations, which admit period mappings to de rham affine grassmannians. after reviewing this theory, we will state a result characterizing the basic admissible pairs that admit cm in terms of transcendence of their periods. this result can be seen as a p-adic analogue of a theorem of cohen and shiga-wolfhart characterizing cm abelian varieties in terms of transcendence of their periods. all work is joint with sean howe. |
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november 2 |
kiran kedlaya (uc san diego)
the fargues-fontaine curve associated to an algebraically closed nonarchimedean field of characteristic $p$ is a fundamental geometric object in $p$-adic hodge theory. via the tilting equivalence it is related to the galois theory of finite extensions of q_p; it also occurs in fargues's program to geometrize the local langlands correspondence for such fields. recently, peter dillery and alex youcis have proposed using a related object, the "affine cone" over the aforementioned curve, to incorporate some recent insights of kaletha into fargues's program. i will summarize what we do and do not yet know, particularly about vector bundles on this and some related spaces (all joint work in progress with dillery and youcis). |
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november 9, 12:40pm |
robin zhang (mit)
the class number formula describes the behavior of the dedekind zeta function at s = 0 and s = 1. the stark and gross conjectures extend the class number formula, describing the behavior of artin l-functions and p-adic l-functions at s = 0 and s = 1 in terms of units. the harris–venkatesh conjecture describes the residue of stark units modulo p, giving a modular analogue to the stark and gross conjectures while also serving as the first verifiable part of the broader conjectures of venkatesh, prasanna, and galatius. in this talk, i will draw an introductory picture, formulate a unified conjecture combining harris–venkatesh and stark for weight one modular forms, and describe the proof of this in the imaginary dihedral case. |
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november 9 |
shou-wu zhang (princeton)
in this talk, we consider a conjecture by gross and kudla that relates the derivatives of triple-product l-functions for three modular forms and the height pairing of the gross—schoen cycles on shimura curves. then, we sketch a proof of a generalization of this conjecture for hilbert modular forms in the spherical case. this is a report of work in progress with xinyi yuan and wei zhang, with help from yifeng liu. |
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november 16, online |
tony feng (uc berkeley)
the breuil-mezard conjecture predicts the existence of hypothetical "breuil-mezard cycles" that should govern congruences between mod p automorphic forms on a reductive group g. most of the progress thus far has been concentrated on the case g = gl_2, which has several special features. i will talk about joint work with bao le hung on a new approach to the breuil-mezard conjecture, which applies for arbitrary groups (and in particular, in arbitrary rank). it is based on the intuition that the breuil-mezard conjecture is analogous to homological mirror symmetry. |
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november 30, apm 7218 and online |
finley mcglade (uc san diego)
the classical spezialschar is the subspace of the space of holomorphic modular forms on $\mathrm{sp}_4(\mathbb{z})$ whose fourier coefficients satisfy a particular system of linear equations. an equivalent characterization of the spezialschar can be obtained by combining work of maass, andrianov, and zagier, whose work identifies the spezialschar in terms of a theta-lift from $\widetilde{\mathrm{sl}_2}$. inspired by work of gan-gross-savin, weissman and pollack have developed a theory of modular forms on the split adjoint group of type d_4. in this setting we describe an analogue of the classical spezialschar, in which fourier coefficients are used to characterize those modular forms which arise as theta lifts from holomorphic forms on $\mathrm{sp}_4(\mathbb{z})$. |
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december 7, apm 7218 |
jon aycock (uc san diego)
the concept of p-adic families of automorphic forms has far reaching applications in number theory. in this talk, we will discuss one of the first examples of such a family, built from the eisenstein series, before allowing this to inform a construction of a family on an exceptional group of type g_2. |